{
  "aliases": [
    "proof-by-contraposition"
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  "body": "\n## Summary\n\nA proof by contrapositive proves an [[implication|0ZNX6DQ3EJ]] $p \\to q$ by a [[direct proof|0NTF6DMVAP]] of $\\neg q \\to \\neg p$.\nThe method is correct because $p \\to q$ is logically equivalent to its [[contrapositive|04T1CT7212]].\n\n## Remarks\n\nExample: for an integer $n$, if $n^2$ is even, then $n$ is even. The contrapositive is \"if $n$ is odd, then $n^2$ is odd\", which has a short direct proof.\nA [[proof by contradiction|0J0XBKG64S]] is different: it assumes $p$ and $\\neg q$ together, and it gets any contradiction.\n\n## Explanations\n\n### L1 human\n\nSometimes the negation of the conclusion gives you more to work with.\n\"If $n^2$ is even, then $n$ is even\" is hard to start.\n\"If $n$ is odd, then $n^2$ is odd\" is easy, and it says the same thing.\n",
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