To prove that ????????, first prove that ????????, then prove that ????????. Example: prove that if ???? is an integer, then ???? is odd if and only if ????^2 is odd. We proved that if ???? is odd, then ????^2 is odd. (????????) We proved that if ????^2 is odd, then ???? is odd. (????????) Therefore, ???? is odd if and only if ????^2 is odd. (????????) To prove that ????_1...????_????, first prove that ????_1????_2, then prove that ????_2????_3, then continue until you have proven that ????_(????1)????_????, then prove ????_????????_1. Example: prove the statements ????_1: ???? is even, ????_2: ????1 is odd, and ????_3: ????^2 is even are equivalent. Show ????_1????_2: if ????=2???? then ????1=2????1=2(????1)+1. Show ????_2????_3: if ????1=2????+1 then ????=2????+2 and ????^2=4????^2+8????+4. Show ????_3????_1: if ???? is odd then ????^2 is odd, so by contraposition if ????^2 is even then ???? is even Explain in the most simple way

. To prove that p q, first prove that p - q, then prove that q - p. . Example: prove that if n is an integer, then n is odd if and only if n2 is odd. 1. We proved that if n is odd, then n2 is odd. (p - q) 2 . We proved that if n is odd, then n is odd. (q - p) 3 . Therefore, n is odd if and only if n2 is odd. (p - q) . To prove that p1 ... Pn, first prove that p1 - P2, then prove that p2 - P3, then continue until you have proven that Pn-1 - Pn, then prove pn - P1. Example: prove the statements p1: n is even, p2: n - 1 is odd, and p3: n is even are equivalent. 1. Show P1 - P2: if n = 2k then n -1 = 2k - 1 = 2(k - 1) + 1. 2 . Show P2 - P3: if n - 1 = 2k + 1 then n = 2k + 2 and n2 = 4k2 + 8k + 4. 3. Show P3 - p1: if n is odd then n2 is odd, so by contraposition if n is even then n is even