The notation a a denotes that a is an element of the

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The notation a A denotes that a is an element of the set A . If a is not a member of A , write a A
Describing a Set: Roster Method S = { a,b,c,d } Order not important S = { a,b,c,d } = { b,c,a,d } Each distinct object is either a member or not; listing more than once does not change the set. S = { a,b,c,d } = { a,b,c,b,c,d } Elipses (…) may be used to describe a set without listing all of the members when the pattern is clear. S = { a,b,c,d, ……,z }
Roster Method Set of all vowels in the English alphabet: V = {a,e,i,o,u} Set of all odd positive integers less than 10 : O = {1,3,5,7,9} Set of all positive integers less than 100 : S = {1,2,3,……..,99} Set of all integers less than 0: S = {…., -3,-2,-1}
Some Important Sets N = natural numbers = {0,1,2,3….} Z = integers = {…,-3,-2,-1,0,1,2,3,…} Z⁺ = positive integers = {1,2,3,…..} R = set of real numbers R + = set of positive real numbers C = set of complex numbers . Q = set of rational numbers
Set-Builder Notation Specify the property or properties that all members must satisfy: S = { x | x is a positive integer less than 100} O = { x | x is an odd positive integer less than 10} O = { x ∈ Z⁺ | x is odd and x < 10} A predicate may be used: S = { x | P( x )} Example: S = { x | Prime( x )} Positive rational numbers : Q + = { x R | x = p / q , for some positive integers p , q }
Universal Set and Empty Set The universal set U is the set containing everything currently under consideration. Sometimes implicit Sometimes explicitly stated. Contents depend on the context. The empty set is the set with no elements. Symbolized ∅, but {} also used. U Venn Diagram a e i o u V John Venn (1834-1923) Cambridge, UK
Some things to remember Sets can be elements of sets. {{1,2,3}, a , { b,c }} {N,Z,Q,R} The empty set is different from a set containing the empty set. ≠ { ∅ }
Set Equality Definition : Two sets are equal if and only if they have the same elements. Therefore if A and B are sets, then A and B are equal if and only if . We write A = B if A and B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}
Subsets Definition : The set A is a subset of B , if and only if every element of A is also an element of B . The notation A B is used to indicate that A is a subset of the set B . A B holds if and only if is true. 1. Because a ∈ ∅ is always false, ∅ ⊆ S ,for every set S . 2. Because a S a S , S S , for every set S .
Showing a Set is or is not a Subset of Another Set Showing that A is a Subset of B : To show that A B , show that if x belongs to A, then x also belongs to B . Showing that A is not a Subset of B : To show that A is not a subset of B , A B , find an element x A with x B . ( Such an x is a counterexample to the claim that x A implies x B .) Examples : 1. The set of all computer science majors at your school is a subset of all students at your school. 2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers.
Another look at Equality of Sets Recall that two sets A and B are equal , denoted by A = B , iff Using logical equivalences we have that A = B iff This is equivalent to A B and B A
Proper Subsets Definition : If A B , but A B , then we say A is a proper subset of B , denoted by A B . If A B , then is true.

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