Let \(a=2k+1\) and \(b=2j+1\) be odd. Which expression shows that \(a+b\) is even?

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Review the concept: Direct proof

A direct proof of an implication \(p\Rightarrow q\) starts from the hypothesis \(p\) and, using definitions and previously proved results, chains steps until it reaches the conclusion \(q\).

More practice problems: direct proof

• ZW-DIRECT-PROOF-001 To prove directly that if n is even then n^2 is even, one starts by writing:
• ZW-DIRECT-PROOF-002 If n=2k+1, what is n^2 equal to?
• ZW-DIRECT-PROOF-004 In a direct proof of p⇒ q, what do we do?
• ZW-DIRECT-PROOF-005 Which of these starts is a direct proof of «if n is odd then n+1 is even»?
• ZW-DIRECT-PROOF-006 Let a=2x and b=2y be even. What follows about ab?
• ZW-DIRECT-PROOF-007 We know a| b and b| c, i.e. b=ak and c=bm. How do we conclude a| c?
• ZW-DIRECT-PROOF-008 Does checking that 3^2=9, 5^2=25 and 7^2=49 are odd prove that the square of every odd number is od…
• ZW-DIRECT-PROOF-009 To prove that if n is odd then 8| n^2-1, write n=2k+1. What is n^2-1 equal to?

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