Discussion #26 Binary Relations: Operations & Properties

Discussion #26 Chapter 5, Sections 3.4-4.5 1/15

Discussion #26

Binary Relations: Operations & Properties

Discussion #26 Chapter 5, Sections 3.4-4.5 2/15

Topics

• Inverse (converse)

• Set operations

• Extensions and restrictions

• Compositions

• Properties– reflexive, irreflexive– symmetric, antisymmetric, asymmetric– transitive

Discussion #26 Chapter 5, Sections 3.4-4.5 3/15

Inverse

• If R: AB, then the inverse of R is R~: BA.

• R-1 is also a common notation

• R~ is defined by {(y,x) | (x,y) R}.– If R = {(a,b), (a,c)}, then R~ = {(b,a), (c,a)}– > is the inverse of < on the reals

• Note that R~ ~R– The complement of < is – But the inverse of < is >

(sometimes called converse)

Discussion #26 Chapter 5, Sections 3.4-4.5 4/15

Set Operations

• Since relations are sets, set operations apply (just like relational algebra).

• The arity must be the same indeed, the sets for the domain space and range space must be the same (just like in relational algebra).

Discussion #26 Chapter 5, Sections 3.4-4.5 5/15

Restriction & Extension• Restriction decreases the domain space or range space.

Example: Let < be the relation {(1,2), (1,3), (2,3)}. The restriction of the domain space to {1} restricts < to {(1,2), (1,3)}.

• Extensions increase the domain space or range space.Example: Let < be the relation {(1,2), (1,3), (2,3)}. The extension of

both the domain space and range space to {1,2,3,4} extends < to {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}

• Extensions and restrictions need not change the relation.– Suppose {M1, M2, M3} are three males and {F1, F2, F3} are three

females and married_to is {(M1,F3), (M2,F2)}.

– Removing the unmarried male M3 or the unmarried female F1 does nothing to the relation, and adding the unmarried male M4 or the unmarried female F4 does nothing to the relation.

Discussion #26 Chapter 5, Sections 3.4-4.5 6/15

12345

1234

1234

1234

Composition• Let R:AB and S:BC be two relations. The composition of R and S,

denoted by RS, contains the pairs (x,z) if and only if there is an intermediate object y such that (x,y) is in R and (y,z) is in S.

• Given R:AB and S:BC, with A = {1,2,3,4}, B = {1,2,3,4,5}, C = {1,2,3,4}, and R = {(1,3), (4,2), (1,1)} and S = {(3,4), (2,1), (4,2)}, we have:

RS = {(1,4), (4,1)}SR = {(3,2), (2,3), (2,1)} (i.e. not commutative)R(SR) = {(1,2), (4,3), (4,1)}(RS)R = {(1,2), (4,3), (4,1)} (i.e. is associative)

R SGraphically:

Discussion #26 Chapter 5, Sections 3.4-4.5 7/15

Composition & Matrix Multiplication

• Relation matrices only contain 0’s and 1’s.• For relational matrix multiplication, use

for *, for +, with 1=T and 0=F.

0001

0000

0000

1000

000104

000003

000002

001011

54321R

00005

00104

10003

00012

00001

4321S

RS(1,1) = (10)(01)(10)(00)(00) = 0

4

3

2

1

4321RS

=

RS = {(1,4), (4,1)}

Discussion #26 Chapter 5, Sections 3.4-4.5 8/15

Combined Operations• All operations we have discussed can be combined

so long as the compatibility requirements are met.• Example:

R = {(1,1), (3,1), (1,2)}S = {(1,1), (2,3)}(R S)~ R

= {(1,1), (3,1), (1,2), (2,3)}~ R= {(1,1), (1,3), (2,1), (3,2)} R= {(1,1), (1,2), (2,1), (2,2)}

R: A B S: B AA = {1,2,3}B = {1,2,3}

Discussion #26 Chapter 5, Sections 3.4-4.5 9/15

Properties of Binary Relations• Properties of binary relations on a set R: AA help

us with lots of things groupings, orderings, …• Because there is only one set A:

– matrices are square |A| |A|– graphs can be drawn with only one set of nodes

R = {(1,3), (4,2), (3,3), (3,4)}

00104

11003

00002

01001

43211

3

2

4

Discussion #26 Chapter 5, Sections 3.4-4.5 10/15

Reflexivity• Reflexive: x(xRx)• Irreflexive: x(xRx)

= is reflexive is irreflexive is reflexive < is irreflexive“is in same major as” is reflexive“is sibling of” is irreflexive“loves” (unfortunately) is not reflexive, neither is it

irreflexive

Reflexive Irreflexive Neither

13

12

11

321

03

02

01

321

13

12

01

321

Discussion #26 Chapter 5, Sections 3.4-4.5 11/15

Symmetry• Symmetric: xy(xRy yRx)• Antisymmetric: xy(xRy yRx x = y)• Asymmetric: xy(xRy yRx)

“sibling” is symmetric“brother_of” is not symmetric, in general, but

symmetric if restricted to males is antisymmetric (if ab and ba, then a=b)< is asymmetric and antisymmetric= is symmetric and antisymmetric is antisymmetric“loves” (unfortunately) is not symmetric, neither

is it antisymmetric nor asymmetric

Discussion #26 Chapter 5, Sections 3.4-4.5 12/15

Symmetry (continued…)

• symmetric:

• antisymmetric:

Symmetric Antisymmetric (Asymmetric too, if no 1’s on the diagonal)

No symmetry properties

113

102

101

321

(always both ways)

(but to self is ok)

(never both ways)

013

12

01

321

113

12

01

321

• asymmetric:(and never to self)

(never both ways)

Discussion #26 Chapter 5, Sections 3.4-4.5 13/15

Transitivity

• Transitive: xyz(xRy yRz xRz)“taller” is transitive< is transitive x<y<z“ancestor” is transitive“brother-in-law” is not transitive

• Transitive: “If I can get there in two, then I can get there in one.”

(for every x, y, z)if and

then

x y z

Discussion #26 Chapter 5, Sections 3.4-4.5 14/15

Transitivity (continued…)• Note: xRy yRz corresponds to xRRz, which is

equivalent to xR×Rz = xR2z; thus we can define transitivity as:

xyz(xR2z xRz)“If I can get there in two, then I can get there in one.”

• For transitivity:– if reachable through an intermediate, then reachable

directly is required.– if (x,z)R2, then (x,z)R

(x,z)R2 (x,z)R R2 R (recall our def. of subset: xA xB, then AB)

Discussion #26 Chapter 5, Sections 3.4-4.5 15/15

Transitivity (continued…)

1003

1002

1101

321

1003

1002

1101

321

1003

1002

1001

321

0003

1002

0101

321

0003

1002

0101

321

0003

0002

1001

321

=

=

R R R2 R

R R R2 R

1

2 3

Not Transitive

Transitive

1

2 3