Example if u is the positive integers less than 100 ie

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Unformatted text preview: in domain D for which P(x) is true. —༉  The truth set of P(x) is denoted by —༉  Example: The truth set of P(x) where the domain is { Mercedes, Toyota, Audi, Nissan, Mazda }, and P(x) is “luxury cars” is the set { Mercedes, Audi } Boolean Algebra —༉  Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. —༉  The operators in set theory are analogous to the corresponding operator in propositional calculus. —༉  Union (disjunction) —༉  Intersection (conjunction) —༉  Complementation (negation) —༉  Etc. —༉  As always there must be a universal set U. All sets are assumed to be subsets of U. Union —༉  Definition: let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set: —༉  Example: what is { 1 , 2 , 3 } ∪ { 3 , 4 , 5 } ? Solution: { 1 , 2 , 3 , 4 , 5 } Venn Diagram for A ∪ B U A B Intersec$on —༉  Definition: the intersection of sets A and B, denoted by A ∩ B, is —༉  Note that if the intersection is empty, then A and B are said to be disjoint. —༉  Example: what is { 1 , 2 , 3 } ∩ { 3 , 4 , 5 } ? Solution: {3} —༉  Example: what is { 1 , 2 , 3 } ∩ { 4 , 5 , 6 } ? Solution: ∅ Venn Diagram for A ∩B U A B Complement Definition: if A is a set, then the complement of the A (with respect to U), denoted by Ā is the set U - A Ā = {x ∈ U | x ∉ A} The complement of A is sometimes denoted by Ac . Example: if U is the positive integers less than 100 (i.e., 1, 2, …, 99), what is the complement of {x | x > 70} Venn Diagram for Complement Solution: {x | x ≤ 70} U Ā A Difference —༉  Definition: let A and B be sets. The difference of A and B, denoted by A – B, is the set containing the elements of A that are not in B. The difference of A and B is also called the complement of B with respect to A. A – B = {x | x ∈ A ∧ x ∉ B} = A ∩!B Venn Diagram for A − B A U B Review Ques$ons Example:...
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This document was uploaded on 02/27/2014 for the course CS 215 at SIU Carbondale.

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