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Mathematics LibreTexts

2.8: Summary

  • Page ID
    62279
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Summary

    • A “two-column proof” is a tool that we use to learn techniques for writing proofs.
      • The left-hand column contains a sequence of assertions.
      • The right-hand column contains a justification for each assertion.
      • Each row of the proof is numbered (in the left margin) for easy reference.
      • A dark horizontal line is drawn to indicate the end of the hypotheses.
      • A dark horizontal line is drawn along the left edge of the proof, and of each subproof.
    • In addition to the basic theorems of , we have two rules that use subproofs:
      • \(\Rightarrow\)-introduction
      • proof by contradiction
    • Proofs often use the Law of Excluded Middle, the Rules of Negation, and contrapositives.
    • Assertions that are in a subproof cannot be used as justification for lines that are not in that same subproof.
    • Writing proofs takes practice, but there are some strategies that can help.
    • Proofs can also be written in English prose, using sentences and paragraphs.
    • To show that a deduction is not valid, find a counterexample.
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