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