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A direct proof establishes the truth of a statement by starting from known facts, definitions, or previously proven results and reasoning forward step by step until the desired conclusion is reached. To prove "If P, then Q" directly, you assume P is true, then use logical steps to show Q must also be true. Each step must follow from previous steps by known rules or definitions — no gaps, no hand-waving, no "it is obvious." Direct proof is the most straightforward proof strategy and the one you should try first.
Start with simple numerical proofs: "Prove that the sum of two even numbers is even." Define even: n is even if n = 2k for some integer k. Let a = 2j and b = 2k. Then a + b = 2j + 2k = 2(j + k), which is 2 times an integer, hence even. Walk through each step explicitly. Then have students write their own proofs for similar claims. Emphasize that every step must be justified — writing "so clearly..." without justification is not acceptable.
A direct proof is the most natural form of mathematical reasoning: you start from what you know and work forward to what you want to show. The strategy is simple in principle — assume the hypothesis, apply definitions and known facts, and arrive at the conclusion. The challenge is in the execution: every step must be justified, and the chain of reasoning must be complete.
Consider a claim like "the sum of two even numbers is even." A direct proof begins by unpacking definitions. What does "even" mean? An integer n is even if n = 2k for some integer k. So let a and b be even: a = 2j and b = 2k for integers j and k. Now compute: a + b = 2j + 2k = 2(j + k). Since j + k is an integer, a + b is 2 times an integer, which means a + b is even. Done.
Notice what happened: you did not check any specific numbers. You did not verify that 2 + 4 = 6 is even, or that 10 + 14 = 24 is even. Instead, you used variables (j and k) that represent any integers, so the proof covers every possible pair of even numbers simultaneously. This is the leap from inductive reasoning (checking cases) to deductive proof (covering all cases at once), and it is exactly what makes the direct proof strategy so powerful.
The structure of a direct proof of "If P, then Q" always follows the same skeleton. Step 1: Assume P. Step 2: Unpack definitions and known facts. Step 3: Reason forward using algebra, logic, or previously proven results. Step 4: Arrive at Q. The proof is complete when you have shown that Q is an unavoidable consequence of P, with no gaps in the reasoning.
A common mistake is to start from the conclusion and work backward. If you are trying to prove "If n is odd, then n² is odd," you should not begin with "n² is odd" and try to derive that n is odd — that would be proving the converse, which is a different statement. Always start from the hypothesis and reason toward the conclusion. There are proof techniques (like proof by contradiction) that start differently, but in a direct proof, the direction is always forward from hypothesis to conclusion.