MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets

MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.


For example: the set of seasons S = {Spring, Summer, Fall, Winter}



element

Each object is called an element of the set


element




element




element





There is a standard notation for indicating the number of elements in a set.




1 2 3 4

The set S above has 4 elements




1 2 3 4

The set S above has 4 elementsso we write




1 2 3 4

The set S above has 4 elements

n(S) = 4so we write


More examples of sets:


More examples of sets:T = {1, 2, 3, 4, 5}


More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}


More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}


More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}



W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
















MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to

definitively determine whether or not any particular object is a member of the set.



A = {1, 2, 3, 4, 5} WELL DEFINEDB= {x : x is tall} NOT WELL DEFINED










MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS

Є is the symbol for “is an element of”



If A = {1, 2, 3, 4, 5}, then 2 Є A.



If A = {1, 2, 3, 4, 5}, then 2 Є A.In general the symbol for “not” something is the symbol for that thing with a diagonal line through it.



If A = {1, 2, 3, 4, 5}, then 2 Є A.In general the symbol for “not” something is the symbol for that thing with a diagonal line through it.

For example, 7 A.


The set containing no elements is calledthe empty set (or sometimes, the null set).



Let M = {x: x is a female U.S. President before 2010}




Because this set is EMPTY, we can write




Because this set is EMPTY, we can write

Ø or { }


The set of all elements under consideration for a particular problem is called the universal set (U).



If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.



If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.If you are looking at course grades in a class where the only

grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.




grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.If you roll a die twice & count how many fives you get

U = {0, 1, 2}.




grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.If you roll a die twice & count how many fives you get

U = {0, 1, 2}.



THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.




For example, if we are showing the results of a coin flip, U = { HEAD , TAIL }









If we roll a single ordinary die, then U =





If we roll a single ordinary die, then U = { 1 , 2 , 3 , 4 , 5 , 6 }


The number of elements in set A is called the cardinal number of the set.



n(A) is read



n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.







If A = { 1 , 2 , 4 , 6 , 8 , 10 }, then n(A) = 6.




A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.





A = { 1 , 2 , 4 }





FINITEA = { 1 , 2 , 4 }





FINITE A = { 2 , 4 , 6 , 8 , …}A = { 1 , 2 , 4 }





FINITEINFINITEA = { 2 , 4 , 6 , 8 , …}A = { 1 , 2 , 4 }

MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS

DEFINITION: EQUALITY OF 2 SETSTwo sets are equal (A = B) if they have exactly the

same elements. Otherwise, we write A ≠ B.




{1, 2, 3, 7} = {1, 7, 3, 2} (Order Is not important.)

Example: {a, b, c} ≠ {a, c, e}












DEFINITION: SUBSETSet A is a subset of set B (written A B)

if every element of A is also an element of B.




{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} is NOT a subset of {dog, pig, goat}
















{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}





/⊆No ‘cat’





/⊆


DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)

if every element of A is also an element of B but A ≠ B.



/



⊆/



/



⊆/



/



⊆/



/



⊆/ Here A = B



/



/⊂ Here A = B



/



/⊂



⊆/



/⊂



/⊂



/⊂No ‘cat’



/⊂



/⊂



/⊂



/⊂IMPORTANT

Pay close attentionto the difference between subset ()

and proper subset().




An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)

Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:

({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.






({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.






({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t

allow for equality.






({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t

allow for equality.






({1, 4, 7} {1, 4, 7} is true)But a proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.






({1, 4, 7} {1, 4, 7} is true)But a proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.

So, both 5 < 5 and {1, 4, 7} {1, 4, 7} are FALSE!


Number of subsets of a set:A set with k elements has subsets.

For the set {a, c, f}, there are = 8 elements.



For the set {a, c, f}, there are = 8 elements.



For the set {a, c, f}, there are = 8 elements.Let’s list them:

Subsets with:0 elements 1 element 2 elements 3 elements

Ø





Ø





Ø {a}, {c}, {f}





Ø {a}, {c}, {f}





Ø {a}, {c}, {f}





Ø {a}, {c}, {f} {a,c},{a,f},{c,f}





Ø {a}, {c}, {f} {a,c},{a,f},{c,f}





Ø {a}, {c}, {f} {a,c},{a,f},{c,f}





Ø {a}, {c}, {f} {a,c},{a,f},{c,f} {a, c, f}


EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same

number of elements…that is, if n(A) = n(B).




Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}





http://www.google.com/url?sa=i&source=images&cd=&cad=rja&docid=nay9GCZRPCg7zM&tbnid=RaAd4Ux5QCxVFM:&ved=0CAgQjRwwAA&url=http://www.teacherfiles.com/clip_art_time_clocks.htm&ei=IgXLUdjHAYia9QS9joCADw&psig=AFQjCNEH9VzszqWqKO4kESNul-Ry3bagnw&ust=1372346018061866









Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}It is in this sense that we say that set A is equivalent to set B.






{a, 2, 5, f} is equivalent to {dog, cat, bird, pig}






{a, 2, 5, f} is equivalent to {dog, cat, bird, pig}

{p, q, x} is NOT equivalent to {a, 2, 5, f}


MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets

Documents

examples of sets

set of seasons s

collection of objects

set builder notation

setfor example

fsdfsdafsdfsdafsd fsd

standard notation

elementeach object