3.4 Truth Tables for the Conditional and the Biconditional

Conditional Statements If the weather is good, then we’ll have a picnic. If we have a triangle, then the sum of the interior angles is 180°. If tomorrow is March 19, I’ll wear something green. I’ll go out with you if you pay for the meal. You must prepare to stop if the traffic light turns orange.

Conditional Statement Forms Person is San Franciscan Person is Californian If person is San Franciscan, then he/she is Californian. If person is Californian, then he/she is San Franciscan. If person is not San Franciscan, then he/she is not Californian . If person is not Californian, he/she is not San Franciscan. p q p → q (conditional) q → p (converse) p’ → q’ (inverse) q’ → p’ (contra- positive)

Your Turn Given p: The waves are high. q: The conditions are good for surfing. State the following in English sentences. p →q (conditional) If the waves are high, then the conditions are good for surfing. q →p (converse) If the conditions are good for surfing, then the waves are high. p’ →q’ (inverse) If the waves are not high, then the conditions are not good for surfing. q’ →p’ (contrapositive) If the conditions are not good for surfing, then the waves are not high.

Truth Table for p  q p  q antecedent  consequent If p then q. A conditional is false only when the antecedent is true and the consequent is false. Conditional p q p  q T T T T F F F T F F If you ask me, then I’ll drive you to the store.

Reason for the Truth Table p  q The professor says: “If you pass the final exam, then you will pass the course.” Case 1: (T, T) You pass the finals, and you pass the course. p T  T: T Case 2: (T, F) You pass the finals, and you fail the course. (F) T  F: F Case 3: (F, T) You fail the finals, and you pass the course. (T) F  T: T Case 4: (F, F) You fail the finals, and you fail the course (T) F  F: T

Truth Table for p → q Statements p and q are causally related p: John eats spinach. q: He gets stronger. p → q: If John eats spinach, then he gets stronger. Statements p and q need not be causally related p: John gets A on the test. q: I’ll be a monkey’s uncle. p → q: If John gets A on the test, then I’ll be a monkey’s uncle.

Truth Tables for p  q Conditional p q p  q T T T T F F F T F F p: John gets A on the test q: I’ll be monkey’s uncle If John gets A, then I’ll be monkey’s uncle. If John gets A, then I will not be monkey’s uncle. If John does not get A, then I’ll be monkey’s uncle. If John does not get A, then I will not be monkey’s uncle Conditional p q p  q T T T T F F F T F F

Truth Table for Converse Construct a truth table for q  p Converse p q q p T T T T F F T F F F

Your Turn p q ~p ~q ~p  ~q T F Construct a truth table for ~p  ~q (inverse of p  q). p q ~p ~q ~p  ~q T F

Truth Table for Contrapositive Construct a truth table for ~q  ~p Contrapositive p q ~q ~p ~q ~p T T F T T F F T F F

Conditional and Its Contrapositive are Equivalent Statements p q ~q ~p p  q q p ~p  ~q ~q ~p T T F T T F F T F F Note the equivalence of p  q and ~q  ~p.

Conditional and Its Contrapositive are Equivalent Statements If you’re cool, you won’t wear clothing with your school name on it. If you wear clothing with your school name on it, you’re not cool. Let p: you’re cool q: you won’t wear clothing with your school name on it. p  q and ~q  ~p are equivalent p q

Conditional Statement and Venn Diagram If you eat spinach, then you will be strong. Let p: you eat spinach q: you will be strong p q p  q q  p ~p  ~q ~q  ~p

Constructing a Truth Table Construct a truth table for [( p  q) ~ p]  q p q p  q ~p (p  q) ~p [( p  q) ~p]  q T T T F T F F T F F

Biconditional Statements p  q p if and only if q: p  q and q  p True only when the component statements have the same value. Truth table for the Biconditional p q p  q q  p p  q T F

Definitions of Symbolic Logic

Determining the Truth Value of a Compound Statement You receive a letter that states that you have been assigned a Super Million Dollar Prize Entry Number -- 665567010. If your number matches the winning pre-selected number and you return the number before the deadline, you will win $1,000,000.00. Suppose that your number does not match the winning pre-selected number, you return the number before the deadline and only win a free issue of a magazine. Under these conditions, can you sue the credit card company for making a false claim?

Determining the Truth Value of a Compound Statement (cont’d) Solution: Assign letters to the simple statements in the claim.

Determining the Truth Value of a Compound Statement (cont’d) Now write the underlined claim in the letter in symbolic form:

Determining the Truth Value of a Compound Statement (cont’d) Substitute the truth values for p, q, and r to determine the truth value for the letter’s claim. (p  q)  r (F  T)  F F  F T Our truth-value analysis indicates that you cannot sue the credit card company for making a false claim.