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Propositional Logic Review Predicate logic Predicate Logic Examples Predicate logic G. Carl Evans University of Illinois Summer 2013 Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Propositional logic AND, OR, T/F, implies, etc Equivalence and truth tables Manipulating propositions Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Implication Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Implication a b T T F F T F T F a→b ¬a ∨ b T F T T a ⇐⇒ b (¬a ∨ b) ∧ (¬b ∨ a) T F F T b T F T F Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Today Be able to incorporate predicates and quantifiers into logical statements Be able to manipulate statements with quantifiers Learn how to prove a universal statement Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Predicate Logic Predicate: propositions that have input variables with a range of values For some integer x, x > 10 Cars that are read and speeding are likely to be ticketed. isred(x) ∧ speeding (x) → likely to be ticketed(x) A person’s mother’s mother is his/her grandmother For every set of people x, y , z mother (x, y ) ∧ mother (y , z) → grandmother (x, z) Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers For some x : ∃x For all x : ∀x For exactly one x : ∃!x Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Binding and Scope ∀x, p(x) → q(x) Binding: ∀x Scope: p(x) → q(x) ∃x, x 2 = 0 Binding: ∃x Scope: x 2 = 0 Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Manipulating quantifiers: Negation Negation: ¬(∀x, p(x)) ≡ ∃x, ¬p(x) ¬(∃x, p(x)) ≡ ∀x, ¬p(x) Examples “Not all dogs are fat” is equivalent to “At least one dog is not fat.” “There does not exist one fat dog” is equivalent to “All dogs are not fat.” Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Contrapositive ∀x, p(x) → q(x) ≡ ∀x, ¬q(x) → ¬p(x) Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a ∀a ∈ Z, ∀b ∈ Z, a + b ≥ a or ∀a, b ∈ Z, a + b ≥ a For every real a, there exists an integer b such that a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a ∀a ∈ Z, ∀b ∈ Z, a + b ≥ a or ∀a, b ∈ Z, a + b ≥ a For every real a, there exists an integer b such that a + b ≥ a ∀a ∈ R, ∃b ∈ Z, a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for some k ∈ Z. Thus ab is odd by definition of odd. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Approach to proving universal statements State the supposition (hypothesis) and define any variables Expand definitions such as “odd” or “rational” into their technical meaning (if necessary) Manipulate expression until claim is verified by a simple statement End with “This is what was to be shown.” or “QED” to make it obvious that the proof is finished. Tip: work out the proof on scratch paper first, then rewrite it in a clear, logical order with justification for each step. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For any real k, if k is rational, then k 2 is rational. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For any real k, if k is rational, then k 2 is rational. Definition: real k is rational iff k = m n for some integers m, n, with n 6= 0. Let k ∈ Q. By definition of rational k = m n k2 = for some m, n ∈ Z with n 6= 0. m 2 m2 = 2 n n Since m, n ∈ Z, m2 , n2 ∈ Z and since n 6= 0, n2 6= 0, k 2 is rational by definition. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For all integers n, 4(n2 + n + 1) − 3n2 is a perfect square. Definition: k is a perfect square iff k = m2 for some integer m. Let n be an integer. 4(n2 + n + 1) − 3n2 = n2 + 4n + 4 = (n + 2)2 Since n is an integer n + 2 is an integer so by definition of perfect square 4(n2 + n + 1) − 3n2 is a perfect square. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: The product of any two rational numbers is a rational number. Definition: real k is rational iff k = m n for some integers m, n, with n 6= 0. Let a, b be rational numbers. By definition of rational a = m n ,b = n and k are not 0. j k for some m, n, j, k ∈ Z s.t. mj mj = nk nk Since m, n, j, k are integers mj, nk are integers and since n and k are not 0 nk 6= 0. Thus by definition of rational ab is rational. QED ab = Predicate logic