Lec1.2 - L1.2 Notes: Logic (continued) Compound...

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L1.2 Notes: Logic (continued) Compound propositions may have specific properties of interest, and some of these are defined here. TAUTOLOGY: A compound proposition that is TRUE for all combinations of its atomic propositions E.g. (p q) (p q) Truth Table: p q p q p q (p q) (p q) T T T T T T F F T T F T F T T F F F F T Note the bold, italicized T’s in the final column. That is the “signature” of a tautology. A tautology is a logical way of combining propositions so that a false result is impossible. It will be the basis of proofs because proofs are intended to ensure that something is true in all circumstances. Another interesting tautology is [(p q) ( ¬ p q)] q This is a symbolic representation of the logic of sentences like “If I do my homework (p) my parents yell at me (q) and if I don’t do my homework my parents yell at me, therefore my parents will yell at me.” 1
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CONTRADICTION: A compound proposition that is FALSE for all combinations of its atomic propositions. Like tautology the term contradiction has a specific meaning in logic and is not to be confused with other commone usages of the terms. E.g. ¬ p (p q) p q p q ¬ p ¬ p (p q) T T T F F T F F F F F T F T F F F F T F Note the bold, italicized F’s in the final column. That is the “signature” of a contradiction. A contradiction cannot be true under any circumstances (i.e., under any assignment of truth values to its atomic propositions). The classic
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Lec1.2 - L1.2 Notes: Logic (continued) Compound...

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