DiscreteMath-Chapter3 - Discrete Math Notes Section 3.1 Proofs Proving a statement is true Must stay GENERIC Even An integer is even if and only if

# DiscreteMath-Chapter3 - Discrete Math Notes Section 3.1...

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Discrete Math NotesMay 23 2011 - Section 3.1Proofs- Proving a statement is true.Must stay GENERIC.Even - An integer is even, if and only if, ∃ “there exists” an integer k, such that, n = 2kOdd - An integer n is odd, if and only if, ∃ an integer k, such that n = 2k+1.Prime- An integer is prime, if and only if, n > 1 and ∀ positive integers r and s, if n = r * s, then r = 1 or s = 1Composite - An integer n is composite, if and only if, n > 1 and ∃ positive int’s r and s, such that n = r*s, and r !=1 and s != 1Proving a “There Exists” statement∃ an even prime numberTRUE- The # 2 is prime and even∃ an integer such that n2= 1 and n != 1TRUE- n = -1Proving a “For All” statementExample 1) ∀ integers a, b - if a and b are both even, then a + b are evenLet a and b be generic but particular integers, such that a and b are both evenBy the definition of even, ∃ an integer k and i, such that a = 2k and b=2i.Substitution: Then a + b = 2 (k + i)As (k + i) is an integer, a + b is even by definition of even.Example 2) The difference between any odd integer and any even integer is oddSuppose n is any odd integer, and m is any even integerBy the definition of odd, ∃ an integer k, such that n = 2k + 1, and By the definition of even, ∃ an integer i, such that m = 2iSubstitution: Then, (n - m) = 2 ( k - i) + 1  • • • 