EXERCISES 3-1. We can use these equivalences to finally do mathematical proofs. To illustrate the logical form of arguments, we use letters of the alphabet (such as p, q, and r) to represent the component sentences of an argument. Examples Find the truth tables for the following statement forms: 1 p_˘q 2 p _(q ^r) 3 (p _q)^(p _r) ... 2.1 Logical Equivalence and Truth Tables 4 / 9. • truth table method and • by the logical proof method (using the tables of logical equivalences.) demonstrated logical equivalence. p q :q p!q :(p!q) p^:q T T F T F F T F T F T T F T F T F F F F T T F F Since the truth values for :(p!q) and p^:qare exactly the same for all possible combinations of truth values of pand q, the two propositions are equivalent. The following tables summarize those rules. See tables 7 and 8 in the text (page 25) for some equivalences with conditionals and biconditionals. Exercise 2: Use truth tables to show that pÙ T ” p (an identity law) is valid. 2 Show that ˘(p _q) ˘p^˘q. A Statement (or Proposition) is a sentence that is true or false but not both. For example: Two plus two equals ve. Also, if you feel you need more practice with truth tables, prove these laws using truth tables. Truth Tables, Tautologies, and Logical Equivalence Mathematics normally works with a two-valued logic : Every statement is either True or False . You can use truth tables to determine the truth or falsity of a complicated statement based on the truth or falsity of its simple components. p q :p p^:q p^q p^:q!p^q T T F F T T T F F T F F F T T F F T F F T F F T j= ’since each interpretation satisfying psisatisﬁes also ’.] Exercise 1: Use truth tables to show that ~ ~p ” p (the double negation law) is valid. Note that all of those rules can be proved using truth tables. Use the truth tables method to determine whether the formula ’: p^:q!p^q is a logical consequence of the formula : :p. Solution. Solution 2. For example: Two plus two equals four. 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