sol2 - MACM 101 — Discrete Mathematics I Outline...

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Unformatted text preview: MACM 101 — Discrete Mathematics I Outline Solutions to Exercises on Predicates, Quantifiers, and Set Theory 1. Determine the truth value of each of these statements if the universe of each variable consists of all real numbers. (a) ∃ x ∀ y ( y negationslash = 0 → xy = 1) ; (b) ∀ x ∃ y ( x + y = 1) . (a) This statement is false. To prove it we have to prove the negation of the statement: ¬ ( ∃ x ∀ y ( y negationslash = 0 → xy = 1) ) ⇐⇒ ∀ x ∃ y ¬ ( y negationslash = 0 → xy = 1) law for negation and quantifiers ⇐⇒ ∀ x ∃ y ¬ ( ¬ ( y negationslash = 0) ∨ ( xy = 1)) expression for implications ⇐⇒ ∀ x ∃ y ( y negationslash = 0 ∧ xy negationslash = 1) DeMorgan’s law. Now take a generic value of x . We need to find a value for y such that ( y negationslash = 0 ∧ xy negationslash = 1) is true. We consider two cases. First, if x = 0 then xy negationslash = 1 for any y . This means that choosing any y negationslash = 0 , say, y = 1 , we make the statement ( y negationslash = 0 ∧ xy negationslash = 1) true. In the second case, x negationslash = 0 , and we choose y = 2 x . Clearly, y negationslash = 0 and xy = 2 negationslash = 1 . (b) This statement is true. Take a generic value of x and find a value of y such that x + y = 1 . Clearly, y = x − 1 satisfies this condition. 2. Find a counterexample, if possible, to this universally quantified statement, where the universe for all variables consists of all integers ∀ x ∀ y ( xy ≥ x ) ....
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sol2 - MACM 101 — Discrete Mathematics I Outline...

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