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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.
 Spring '14
 M.Nojoumian
 Computer Science

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