Ch 1 2 Conditionals Biconditionals Definitions Given propositions

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. P Q ~P ~P / Q P =>Q T T F T T T F F F F T T F F T

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

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) P Q P =>Q T T T F F F F T

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.

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. P Q ~P ~Q P =>Q Q =>P ~Q =>~P T T F F T T T F T F T T F F F T T T

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 P Q T T T F T F F T

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.

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.

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. ”

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)

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