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Mathematics LibreTexts

1.10: Summary

  • Page ID
    62090
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    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


    This page titled 1.10: Summary is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dave Witte Morris & Joy Morris.

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