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Unformatted text preview: ans substitute term t for all variables x in A. • Sxt A is called an instantiation of A and t is said to be an instance of x. • Examples: Sx3 P(x, y) = P(3, y) Sx3+1 P(x, y, z, x) = P(3+1, y, z, 3+1) = P(4, y, z, 4) Discussion #14 Chapter 2, Section 1 10/20 Universal Quantification • Let A be an expression, and let x be a variable. If we want to say that P(x) is true for all substitutions of values for x in the UofD, we write ∀xP(x). • The symbol ∀ is pronounced “for all” and is called the universal quantifier. • Examples: – All cats have tails, ∀x(cat(x) ⇒ hasTail(x)). – For every integer x, x+1 > x, ∀x(>(x+1, x)). Discussion #14 Chapter 2, Section 1 11/20 Universal Quantification (continued…) ∀ ∀x P(x) is shorthand for: ∀x P(x) = P(a) ∧ P(b) ∧ P(c) with UoD = {a, b, c}. ∀x P(x) = P(0) ∧ P(1) ∧ … with UoD = non­negative integers. ∀ ∀x P(x) = T when P(x) = T for all substitutions from the UoD. (Only need one false predicate instantiation to make the formula false.) • Examples: ∀x red(x) = T for UoD = red apples ∀x red(x) = F for UoD = apples Discussion #14 Chapter 2, Section 1 12/20 Existential Quantification • Let A be an expression, and let x be a variable. If we want to say that P(x) is true for at least one value of x, we write ∃ xP(x). • The symbol ∃ is pronounced “there exists” and is called the existential quantifier. • Examples: – Some people like apples, ∃ x(likesApples(x)). – There is an integer larger than 10, ∃ x(>(x, 10)). Discussion #14 Chapter 2, Section 1 13/20 Existential Quantification (continued…) ∀ ∃ x is shorthand for: ∃ x P(x) = P(a) ∨ P(b) ∨ P(c) with UoD = {a, b, c}. ∃ x P(x) = P(1) ∨ P(2) ∨...
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