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Notation s t indicates that s and t are logically

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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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