13.3 Truth Tables for the Conditional and the Biconditional

Transcription

1 ruthablesconditionalbiconditional.nb ruth ables for the Conditional and the Biconditional Conditional Earlier, we mentioned that the statement preceding the conditional symbol is called the antecedent and that the statement following the conditional symbol is called the consequent. or example, consider (p fi q) ö [~(q fl r)]. In this statement, (p fi q) is the antecedent and [~(q fl r)] is the consequent. Now we look at truth tables for the conditional. Consider the statement "If you get an A in class, then I will make you cookies." Assume this statement is true except when I have actually broken my promise to you. Let p: You get an A. q: I make you cookies. ranslated into symbolic form, the statement becomes p ö q. Let's examine the four cases as shown in the table below. Conditional p q p ö q Case 1: (, ) You get an A, and I make you cookies. I have met my commitment and the statement is true. Case 2: (, ) You get an A, and I do not make you cookies. I have broken my promise, and the statement is false. Case 3: (, ) You do not get an A, and I make you cookies. I have not broken my promise, and therefore the statement is true.

2 ruthablesconditionalbiconditional.nb 2 Case 4: (, ) You do not get an A, and I don't make you cookies. I have not broken my promise, and therefore the statement is true. he conditional statement is false when the antecedent is true and the consequent is false. In every other case the conditional statement is true. à he conditional statement p ö q is true in every case except when p is a true statement and q is a false statement. Example 1: A ruth able with a Conditional. Construct a truth table for the statement p ö ~q. Solution: (You have to do the work here. I am getting tired of typing) he table is partially filled out. ill in the columns in the appropriate order. Check to make sure your table is correct. p q p ö ~q Example 2: A Conditional ruth able with hree Simple Statements Construct a truth table for the statement p ö (q fl ~r) Solution: Same deal as above. Complete the table below following the dominance of connectives and parentheses. Remember to number your columns.

3 ruthablesconditionalbiconditional.nb 3 p q r p ö Hq ö ~rl Biconditional he biconditional statement, p õ q, means p ö q and q ö p, or symbolically, (p ö q) fl (q ö p). Complete the table below for this conjunction. p q Hp ö ql fl Hq ö pl Now we can construct a truth table for the biconditional p õ q from the results of the table above.

4 ruthablesconditionalbiconditional.nb 4 Biconditional p q p õ q à he biconditional statement, p õ q, is true only when both p and q have the same truth value, that is, when both are true or both are false. Example 3: A ruth able Using a Biconditional Construct a truth table for the statement p õ (q ö ~r) Solution: Again, you have to do the work (sorry!) p q r p õ Hq ö rl Example 4: Determine the ruth Value of a Compound Statement Determine the truth value of the statement (q õ r) ö (~p fl r) when p is true, q is false, and r is true.

5 ruthablesconditionalbiconditional.nb 5 Solution: We do not have to construct a truth table because we are only concerned with one case; where p is true, q is false, and r is true. Just substitute the truth value for each simple statement. (q õ r) ö (~p fl r) ( õ) ö ( fl ) ö Self-Contradictions, autologies, and Implications wo special situations can occur in the truth table of the compound statement: he statement may always be true, or the statement may always be false. We give such statements special names. à A self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is false, the statement is a self-contradiction. Example 4: All alses, a Self-Contradiction Construct a truth table for the statement (p õ q) fl (p õ ~q). Solution: Complete the table below.

6 ruthablesconditionalbiconditional.nb 6 p q Hp õ ql fl Hp õ ~ql Is every truth value in the answer column false? (they should be). his is an example of a self-contradiction. à A tautology is a compound statement that is always true. When every truth value in the answer column of the truth table is true, the statement is a tautology. Example 5: All rues, a autology Construct a truth table for the statement (p fl q) ö (p fi r). Solution: he answer is given in column three of the table below. he truth values are true in every case. hs, the statement is an example of a tautlogy or a logically true statement.

7 ruthablesconditionalbiconditional.nb 7 p q r Hp fl ql ö Hp fi ql he conditional statement (p fl q) ö (p fi r) is a tautology. Conditional statements that are tautologies are called implications. In example 5, we can state p fl q implies p fi r. à An implication is a conditional statement that is a tautology. In any implication the antecedent of the conditional statement implies the consequent. In other words, if the antecedent is true, then the consequent must also be true. hat is, the consequent will be true whenever the antecedent is true. his is why we read the conditional statement p ö q as "If p, then q" instead of "p implies q" as in other texts. We see that "p implies q" only when the conditional is an implication. Example 6: An Implication? Determine whether the conditional statement [(p fl q) fl p] ö q is an implication. Solution: If the conditional statement is a tautology, the conditional statement is an implication. Since the conditional statement is a tautology (see the truth table below), the conditional statement is an implication.

8 ruthablesconditionalbiconditional.nb 8 p fl ql fl pd ö q