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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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