Understanding Propositional Equivalences and Logical Equivalences
Explore the concepts of tautologies, contradictions, and contingencies in propositional logic, along with examples demonstrating logical equivalences. Learn how to show logical equivalence between compound propositions by analyzing truth tables and applying negations.
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Presentation Transcript
1.2 Propositional Equivalences DEFINITION 1 A compound proposition that is always true, no matter what the truth values of the propositions that occur in it, is called a tautology. A compound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency. 1 L Al-zaid Math1111
EXAMPLE 1 We can construct examples of tautologies and contradictions using just one propositional variable. Consider the truth tables of p v -p and p -p, shown in Table 1 . Because p v -p is always true, it is a tautology. Because p -p is always false, it is a contradiction. 2 L Al-zaid Math1111
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Logical Equivalences DEFINITION 2 The compound propositions p and q are called logically equivalent if p q is a tautology. The notation p q denotes that p and q are logically equivalent. 4 L Al-zaid Math1111
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EXAMPLE 2 Show that - (p v q ) and -p -q are logically equivalent. 6 L Al-zaid Math1111
EXAMPLE 3 Show that p q and -p v q are logically equivalent. 7 L Al-zaid Math1111
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Homework Page 34,35 1(b,c) 2 9 (a,e) 16 9 L Al-zaid Math1111
