# The Foundations: Logic and Proofs

Set Operations Union Definition: The union of sets A and B, denoted by A B, contains those elements that are in A or B or both: Example: {1, 2, 3} {3, 4, 5} = {1, 2, 3, 4, 5} U A Venn Diagram for A B B Intersection Definition: The intersection of sets A and B,

denoted by A B, contains elements that are in both A and B If the intersection is empty, then A and B are disjoint. Examples: {1, 2, 3} {3, 4, 5} = {3} {1, 2, 3} {4, 5, 6} = Venn Diagram for A B U A B Difference Definition: The difference of sets A and B, denoted by AB, is the set containing the

elements of A that are not in B: AB = {x | x A x B} Example: {1, 2, 3} {3, 4, 5} = {1, 2} U A B Venn Diagram for AB Complement Definition: The complement of a set A, denoted by is the set UA; i.e., it contains all elements that are not in A = {x U | x A} Example: If U are positive integers less than 100 then the complement of {x | x > 70} is {x | x 70}

U A Venn Diagram for Complement Examples U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {1, 2, 3, 4, 5} B ={4, 5, 6, 7, 8} A B = {1, 2, 3, 4, 5, 6, 7, 8} A B = {4, 5} = {0, 6, 7, 8, 9, 10} = {0, 1, 2, 3, 9, 10} AB = {1, 2, 3} BA = {6, 7, 8} Set Identities

How can we compare sets constructed using the various operators? Identity laws Domination laws Idempotent laws Complementation law Set Identities Commutative laws Associative laws Distributive laws Set Identities De Morgans laws Absorption laws

Complement laws Proving Set Identities Different ways to prove set identities: 1. Prove that each set is a subset of the other. 2. Use set builder notation and propositional logic. 3. Membership Tables: To compare two sets S1 and S2, each constructed from some base sets using intersections, unions, differences, and

complements: Consider an arbitrary element x from U and use 1 or 0 to represent its presence or absence in a given set Construct all possible combinations of memberships of x in the base sets Use the definitions of set operators to establish the membership of x in S1 and S2 S1=S2 iff the memberships are identical for all combinations Membership Table Exampl Show that the distributive law holds. e: Solution: A 1 1 1 1

0 0 0 0 B 1 1 0 0 1 1 0 0 C 1 0 1

0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 1 1

1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1

0 1 0 1 1 1 1 1 0 0 0

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