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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 1.2 Logical Equivalence Consider the following two compound statements 7 is an odd integer and 6 is an even integer. and 6 is an even integer and 7 is an odd integer. If we define statement variables: p : 7 is an odd integer. q : 6 is an even integer. then the first statement has symbolic form p q while the second has symbolic form q p . Our common sense, however, tells us that these two statements are really stating exactly the same thing and hence will have the same truth value, which in this case is true. Furthermore, for any simple statements p and q , the compound statement p q will be true except when one of p or q is false. Similarly, the compound statement q p will also be true except when one of q or p is false. It follows that the two compound statement forms p q and q p will always have the same truth value. Statement forms for which this is the case, that is, statement forms that always have the same truth value are said to be logically equivalent . Definition. Two compound statement forms that have the same truth values for all possible choices of truth values of their statement variables are said to be logically equivalent . We denote the logical equivalence of two compound statement forms P and Q by P Q . To verify that two statement forms P and Q are logically equivalent, we need to show that they have the same truth values for all choices of truth values of their statement variables. In order to show this, we can construct the truth table for each and check that the truth value of P is the same as the truth value of Q in each row of the tables. Example. Show that ( p ) p . This shows the negation of the negation of a statement is logically equivalent to the original statement. Solution. The truth table for ( p ) is as follows. p p ( p ) T F T F T F We see from the table that p and ( p ) have the same truth value in each row of the table. Therefore ( p ) p . According to the logical equivalence in the preceding example, if we need to negate a statement which already contains a negation, such as 7 is not an odd integer. then the resulting statement It is not the case that 7 is not an odd integer. is equivalent to the much simpler statement 7 is an odd integer. Writing simplified versions of negated statements is one of the most important applications of logical equivalences. Well see below how to simplify negations of any compound statement. 13 14 Chapter 1 Propositional Logic Example. Determine whether or not the following statement forms are logically equivalent. (a) ( p q ) and ( p ) ( q ) Solution. For this example, we need to construct the truth tables for the two compound statement forms ( p q ) and ( p ) ( q ). We can save ourselves some writing by combining the two tables into a single table, where the combinations of truth values for the statement variables...
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This note was uploaded on 10/20/2009 for the course MATH 280 taught by Professor Solheid during the Spring '08 term at CSU Fullerton.
- Spring '08