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Unformatted text preview: denoting “x is mortal”. Specify the domain as all people. ༉ The two premises are: ༉ The conclusion is: ༉ Later we will show how to prove that the conclusion follows from the premises. Equivalences in Predicate Logic ༉ Statements involving predicates and quantiﬁers are logically equivalent if and only if they have the same truth value ༉ for every predicate substituted into these statements and ༉ for every domain of discourse used for the variables. ༉ Notation S ≡T indicates that S and T are logically equivalent. ༉ Example: ∀x ¬¬S(x) ≡ ∀x S(x) Quan$ﬁers: Conjunc$ons and Disjunc$ons ༉ If the domain is ﬁnite: ༉ A universally quantiﬁed proposition is equivalent to a conjunction of propositions without quantiﬁers. ༉ An existentially quantiﬁed proposition is equivalent to a disjunction of propositions without quantiﬁers. ༉ If domain consists of the integers 1,2, and 3: ༉ Even if the domains are inﬁnite, you can still think of the quantiﬁers in this fashion, but the equivalent expressions without quantiﬁers will be inﬁnitely long. Nega$ng Quan$ﬁed Expressions ༉ Consider ∀x J(x) “Every student in your class has taken a course in Java.” Here J(x) is “x has taken a course in Java” and the domain is students in your class. ༉ Negating the original statement gives “It is not the case that every student in your class has taken Java.” ༉ This implies that “There is a student in your...
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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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