First proofs and counterexamples

• Most beginner proofs follow a predictable structure: start with an arbitrary object, rewrite assumptions using definitions, then manipulate until you reach the conclusion. For example, to prove "if n is even then n^2 is even," you begin by letting n be even, which by definition means n = 2k for some integer k. That definition is powerful because it turns a word ("even") into algebra you can use. Once you rewrite in definitional form, the proof often becomes straightforward algebra: n^2 = (2k)^2 = 4k^2 = 2(2k^2), which matches the definition of even again.
• Disproving is different: to refute a "for all" claim, you only need one counterexample, and the best counterexamples are simple and decisive. If someone claims "all primes are odd," the number 2 instantly kills it. If someone claims "for all integers n, n^2 > n," then n = 0 or n = 1 refutes it. Learning when to prove and when to counterexample is a core skill: your first step should always be identifying whether a statement is universal (needs proof, vulnerable to counterexample) or existential (needs an example or construction).

• If n is even then n^2 is even (direct proof with n = 2k).
• Disprove 'for all integers n, n^2 > n' with n = 0.

• 1Prove: If n is odd, then n^2 is odd.
• 2Disprove: For all integers n, 2n + 1 is even.
• 3Prove: For all real numbers x, x^2 >= 0.

• 1Translate: 'Every integer has an additive inverse.'
• 2Negate: 'There exists a real x such that x^2 < 0.'
• 3Give a counterexample to: 'All primes are odd.'

MCQ

Which is a valid counterexample to 'For all integers n, n^2 > n'?