OPPA European Social Fund Prague & EU: We invest in your ... · AE4M33RZN,Fuzzylogic: Tutorialexamples RadomírČernoch radomir.cernoch@fel.cvut.cz FacultyofElectricalEngineering,CTUinPrague
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OPPA European Social FundPrague amp EU We invest in your future
AE4M33RZN Fuzzy logicTutorial examples
Radomiacuter Černochradomircernochfelcvutcz
Faculty of Electrical Engineering CTU in Prague
2012
Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
AE4M33RZN Fuzzy logicTutorial examples
Radomiacuter Černochradomircernochfelcvutcz
Faculty of Electrical Engineering CTU in Prague
2012
Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 2
AssignmentOn the universeΔ = a b c d there is a fuzzy set
120583A = 1114106(a 03) (b 1) (c 05)1114109
Find its horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
X 120572 = 0a b c 120572 isin (0 03⟩b c 120572 isin (03 05⟩b 120572 isin (05 1⟩
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 3
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
a b c d 120572 isin ⟨0 13⟩a d 120572 isin (13 12⟩d 120572 isin (12 23⟩empty 120572 isin (23 1⟩
Find the vertical representation
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Solution
120583A(a) = sup1114106120572 isin ⟨0 1⟩ ∶ a isin 120449A(120572)1114109 = sup⟨0 12⟩ = 12120583A(b) = 13 120583A(c) = 13 120583A(d) = 23 therefore
120583A = 1114106(a 12) (b 13) (c 13) (d 23)1114109
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 5
AssignmentOn the universeΔ = IR there is a fuzzy set A
120583A(x) =
⎧⎪⎪⎨⎪⎪⎩
x x isin ⟨0 1⟩2 minus x x isin (1 15⟩0 otherwise
Find its horizontal represenation
Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Solution
120449A(120572) =
⎧⎪⎪⎨⎪⎪⎩
IR 120572 = 0
⟨120572 15⟩ 120572 isin (0 05⟩⟨120572 2 minus 120572⟩ 120572 isin (05 1⟩
Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 7
AssignmentThe fuzzy set A has a horizontal representation
120449A(120572) =⎧⎪⎨⎪⎩
IR 120572 = 0
⟨1205722 1) otherwise
Find the vertical representation
120583A(x) =⎧⎪⎨⎪⎩
radicx x isin (0 1)0 otherwise
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 8
AssignmentDecide if the following function is a fuzzy conjunction
120572and∘ 120573 =
⎧⎪⎪⎨⎪⎪⎩
120572 120573 = 1120573 120572 = 1120572120573 120572120573 ge 110 max(120572 120573) lt 10 otherwise
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Solution
The first two possibilities ensure the boundary condition Comutativityand monotonicity are trivially satisfied If one of the arguments is 1 theassociativity as well For 120572 120573 120574 lt 1 the associativity follows from
120572and∘ 1114102120573and∘ 1205741114105 = 1114108120572120573120574 120572120573120574 ge 1100 otherwise
We get the same for 1114102120572and∘ 1205731114105and∘ 120574
It is always a fuzzy conjunction (interpretable as an algebraicconjunction in which we ignore small values eg for filtering smallvalues)
Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 10
AssignmentDecide if the following function is a fuzzy conjunction
120572 ⋄ 120573 = 1114108120572120573 120572120573 ge 001 or max(120572 120573) = 10 otherwise
Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 11
AssignmentDecide if the following function is a fuzzy conjunctionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
min(120572 120573) 120572 + 120573 ge 1
0 otherwise
Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 12
AssignmentDecide if for all 120572 120573 isin [0 1] holds (120572 ∘or 120573)and
1113693(120572 ∘ornot
1113700120573) = 120572 where the
disjunction∘or is
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 13
AssignmentDecide if the functionand∘ ∶ ⟨0 1⟩
2 rarr ⟨0 1⟩
120572and∘ 120573 =⎧⎪⎨⎪⎩
120572120573 120572 + 120573 ge 1
0 otherwise
is a fuzzy conjunction
Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 14
AssignmentDecide if the (120572 and 120572) or (120572 and 120572) le 120572 holds for
1 standard1113700or
2 algebraic1113682or
3 Łukasiewicz 1113693or
Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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Task 15
AssignmentDecide which equalities hold
1 (120572and1113700120572) 1113693or(120572and
1113700120573) = 120572and
1113700(120572 1113693or120573)
2 (120572and1113693120572) 1113700or(120572and
1113693120573) = 120572and
1113693(120572 1113700or120573)
3 120572 1113700or(120572and1113693120573) = 120572and
1113693(120572 1113700or120573)
Justify your conclusions
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 16
AssignmentDecide which equalities hold
1 (120572and1113700120573) and
1113693120574 = 120572and
1113700(120573 and
1113693120574)
2 not1113700(120572 1113682or120573) = not
1113700120572and
1113693not1113700120573
3 (120572and1113693120572) 1113693ornot
1113700120572 = (not
1113700120572and
1113693not1113700120572) 1113693or120572
Justify your conclusions
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 17
AssignmentVerify that 120572and∘ (120572
1113699rArr∘ 120573) = 120572and1113700120573 holds for
1 algebraic ops
2 standard ops
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Solution
We will show the solution for algebraic operations The other ones aresimilar
120572and1113682(120572 1113699rArr
1113682120573) =
⎧⎪⎨⎪⎩
120572 sdot 1 = 120572 for 120572 le 120573
120572 sdot 120573120572 = 120573 for 120572 gt 120573
⎫⎪⎬⎪⎭= min(120572 120573) = 120572and
1113700120573
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Task 19
AssignmentComplete the table so that R is a S-partial order
R a b c d
ab 05c 03d 02
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
OPPA European Social FundPrague amp EU We invest in your future
Solution
Reflexivity implies 1rsquos on the diagonal S-partial order implies 0rsquos tonon-zero elements symmetric over the main diagonal
R a b c d
a 1 0 xprime yprime
b 05 1 0 0c x 03 1 zprime
d y 02 z 1
The transitivity implies eg R(3 2) and1113700R(2 1) le R(3 1) which translates
into a conditionmin(03 05) le xUsing this and similar conditions we derive the subspace of allsolutions z le 02 x ge 03 y ge 02min(y zprime) le a xprime = yprime = 0
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