Direct proof and contraposition

Suppose an integer is even. This means there exists an integer k with n equal to twice k. Writing that witness turns a descriptive word into a form that algebra can use. If two integers are even, introduce separate witnesses for them; using the same letter can accidentally impose equality between unrelated inputs. To show that the sum is even, factor out two and explicitly note that the remaining expression is an integer, supplying the witness required by the conclusion.

A proof must establish the requested property, rather than stop at an attractive formula. For divisibility, the conclusion requires an integer multiplier. A real multiplier would be useless because any nonzero real divisor divides every real number in that looser sense. State closure facts when they carry the argument: sums and products of integers are integers. With practice those justifications become short, but they remain the reason that algebraic expressions answer a number-theoretic question.

Exploration often starts at the desired conclusion and works backward. This can reveal the calculation you need, but a finished direct proof must show that the hypothesis actually implies every required step. Reversing an irreversible operation is dangerous. Squaring an equation may create additional solutions, and dividing by a variable may discard a case. Label such transformations as necessary conditions during exploration, then either justify their reversal or verify resulting candidates in the original problem.

A useful drafting technique is to write the hypothesis on one side of the page and the definition of the goal on the other. Ask what missing bridge would connect them. To prove a number odd, seek a representation twice an integer plus one. To prove set inclusion, take an arbitrary element of the first set and establish membership in the second. The proof method supplies a route, while the relevant definitions specify exactly what arrival at the goal looks like.

The implication P implies Q is equivalent to “not Q implies not P.” Over the integers, proving that an even square has an even base is easier through its contrapositive: an odd base has an odd square. Write an odd integer as twice an integer plus one, expand the square, and exhibit the result in the same form. This uses the exhaustive even-or-odd classification of integers; the negation of evenness becomes an explicit odd representation.

Contrast this with the inverse “not P implies not Q,” which is equivalent to the converse rather than the original implication. A common mistake is to prove that an even base has an even square when asked to prove that an even square has an even base. Both happen to be true, but one does not establish the other by logic alone. Before a contrapositive proof, name P and Q and write their precise negations, including the domain.