Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.10: Summary

  • Page ID
    62090
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Summary

    • Assertions stated in English can be translated into (and vice-versa)
    • In mathematics, “or” is inclusive.
    • Notation:
      • \(\lnot\) (not; means “It is not the case that”)
      • \(\&\) (and; means “Both ______ and ______”)
      • \(\lor\) (or; means “Either ______ or ______”)
      • \(\Rightarrow\) (implies; means “If ______ then ______”)
      • \(\Leftrightarrow\) (iff; means “______ if and only if ______”)
    • Important definitions:
      • assertion
      • deduction
      • valid, invalid
      • tautology
      • contradiction
      • logically equivalent
      • converse
      • contrapositive
    • Determining whether an assertion is true (for particular values of its variables)
    • An implication might not be equivalent to its converse.
    • Every implication is logically equivalent to its contrapositive.
    • Basic laws of :
      • Law of Excluded Middle
      • rules of negation
      • commutativity of \(\&\), \(\lor\), and \(\Leftrightarrow\)
      • associativity of \(\&\) and \(\lor\)
    • “Theorem” is another word for “valid deduction”
    • Basic theorems of :
      • repeat
      • introduction and elimination rules for \(\&\), \(\lor\), and \(\Leftrightarrow\)
      • elimination rule for \(\Rightarrow\)
      • proof by cases

    • Was this article helpful?