Ch 1.2: Conditionals & Biconditionals

1. Ch 1.2: Conditionals & Biconditionals • Definitions: Given propositions P and Q, • The conditional sentence P => Q (read “P implies Q”) is the proposition “If P, then Q.” • antecedent = P, consequent = Q • P => Q is true whenever the antecedent is false or the consequent is true, so P => Q is defined to be equivalent to (~P) \/ Q.

2. Conditional P => Q • If it rains tomorrow, then I will give you a ride to the store.

3. Conditional P => Q • Note that if both P and P => Q are true, then according to the truth table, Q is also true. • This deduction is called modus ponens (Ch 1.4)

4. Converse & Contrapositive • Definitions: For propositions P and Q, • The converse of P => Q is Q => P • The contrapositive of P => Q is ~Q => ~P • Example: If I am hungry, then I will eat lunch. • Converse: If I am eating lunch, then I am hungry. • Contrapositive: If I am not eating lunch, then I am not hungry.

5. Converse & Contrapositive • Theorem 1.1: For propositions P and Q, • P => Q is not equivalent to its converse Q => P • P => Q is equivalent to its contrapositive ~Q => ~P • Proof: From the truth table below, we see that P => Q is not equivalent to Q => P and is equivalent to ~Q => ~P.

6. Biconditional • Definition: For propositions P and Q, • The biconditional sentence PQ is the proposition “P if and only if Q.” • The sentence PQ is true exactly when P and Q have the same truth values (T or F). • Examples • “I am eating iff I am hungry” • “2+3=4 iff Denver is in Arizona” • “2+3=5 iff Pi is rational” • x^2 – 4 = 0 iff (x-2)(x+2)=0

7. Theorem 1.2 • Theorem 1.2: For propositions P, Q, and R, • P => Q is equivalent to (~P) \/ Q • P Q is equivalent to (P => Q) /\ (Q =>P) • ~(P /\ Q) is equivalent to (~P) \/ (~Q) • ~(P \/ Q) is equivalent to (~P) /\ (~Q) • ~(P => Q) is equivalent to P /\ (~Q) • ~(P /\ Q) is equivalent to P => ~Q • P /\ (Q \/ R) is equivalent to (P /\ Q) \/ (P /\ R) • P \/ (Q /\ R) is equivalent to (P \/ Q) /\ (P \/ R) • Proof: Use previous results and truth tables to establish these results.

8. English Translation • Use P => Q for the following: • If P, then Q • P implies Q • P is sufficient for Q • P only if Q • Q, if P • Q whenever P • Q is necessary for P • Q, when P • Example: Run through the above for P = differentiability of f, Q=continuity of f.

9. English Translation • Use P  Q for the following: • P if and only if Q • P if, but only if Q • P is equivalent to Q • P is necessary and sufficient for Q • Example: Run through the above for P = “x^2-4 = 0,” Q=“(x-2)(x+2)=0.”

10. English Translation • Example: Write using logical connectives: • A number x is real and not rational whenever x is irrational • (x irrational) => [(x is real) /\ ~(x is rational)] • Example: Write using logical connectives: • A sequence x in R is Cauchy iff x is convergent. • (x a sequence in R)  (x is Cauchy)

11. Homework • Read Ch 1.2 • Do 15(1,2,4a-e,5a-e,6a-d,8a,b,e,13)