Mathematical proof and notation
Objective. Use implication, equivalence, exhaustion and counterexample with precise notation.
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- 2. Worked example
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Learn: the key idea
Mathematical proof is a chain of logical steps that establishes a statement is always true, using precise symbols for implication (⇒), equivalence (⇔) and "there exists". Proof by deduction builds from known facts; proof by exhaustion checks every case in a finite list; disproof by counterexample needs only one failing case. For example, to disprove "all primes are odd" the single counterexample 2 suffices, since 2 is prime and even. Always state which method you are using and keep each line justified so the argument reads as a complete, checkable sequence.
Worked example
Give a proof by exhaustion that n²+n is even for every integer n with 0 ≤ n ≤ 3.
- 1List the cases: n = 0, 1, 2, 3
- 2Compute n²+n for each: 0, 2, 6, 12
- 3Each value divides exactly by 2 with no remainder
- 4All four cases are even, so the statement holds for 0 ≤ n ≤ 3
Answer: n²+n is even for n = 0, 1, 2, 3
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Start practisingCommon mistake. Students write ⇒ when they mean ⇔, claiming a one-way deduction also works in reverse; for instance x²=4 ⇒ x=2 is false since x could be −2, so equivalence must be checked both ways before using ⇔.
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