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直接証明とは 方法 of showing the truth of a given statement by a straightforward combination of established facts, usually axioms, definitions, and previously proven theorems. To prove a conditional statement [latex]p \rightarrow q[/latex], one assumes that [latex]p[/latex] is true and uses rules of inference to show that [latex]q[/latex] しなければならない also be true.
A direct proof is the most straightforward form of proof in mathematics. It proceeds logically from a set of premises or hypotheses to a conclusion. The structure typically follows a linear chain of reasoning. For instance, to prove the statement “If n is an odd integer, then n² is an odd integer,” a direct proof would start by assuming n is odd. By definition, an odd integer can be written as [latex]n = 2k + 1[/latex] for some integer k. Squaring this expression gives [latex]n^2 = (2k + 1)^2 = 4k^2 + 4k + 1[/latex]. This can be rewritten as [latex]n^2 = 2(2k^2 + 2k) + 1[/latex]. Since [latex]2k^2 + 2k[/latex] is an integer, let’s call it m, we have [latex]n^2 = 2m + 1[/latex]. By definition, this shows that n² is an odd integer. This example illustrates the core principle: using definitions and algebraic manipulation to directly arrive at the desired conclusion from the initial assumption. This method is fundamental and forms the basis of many more complex proofs. It relies on the law of detachment (modus ponens) in logic.
直接証明（数学）
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