Logic ChAPTER 3

The Conditional and the Biconditional 3.3

Truth Value of the Conditional Statement The conditional statement p → q (“if p then q”) is false only when p is true and q is false; otherwise, it is true. In p → q, statement p is called the antecedent and q the consequent.. p q p → q T F

Examples Find the truth value of each statement. a. If 2+2 = 22, then 22 = 4. b. If 2+2 = 4, then 8 = 5. c. If 2+2 = 22, then 8 = 4+4. TRUE FALSE TRUE

Reason: a. If 2+2 = 22, then 22 = 4. p: 2+2 = 22 q: 22 = 4 FALSE p → q ; F → F is T FALSE b. If 2+2 = 4, then 8 = 5. p: 2+2 = 4 q: 8 = 5 TRUE p → q ; T → F is F FALSE c. If 2+2 = 22, then 8 = 4+4. p: 2+2 = 22 q: 8 = 4+4 FALSE p → q ; F → T is T TRUE

Examples Find all the number replacements for x that make each sentence true. d. If 2 + 2 = 4, then x – 2 = 5. e. If 2 + 2 = 22, then x – 2 = 5. f. If x + 2 = 6, then 3 + 2 = 5. x = 7 x = any number x = any number

Reason: d. If 2+2=4, then x – 2 = 5. p: 2+2 = 4 q: x – 2 = 5 TRUE Refer to the truth table on slide three. Since p is true q must be true to have a true conclusion. Hence, x – 2 = 5 must be solved algebraically. x – 2 = 5 + 2 + 2 x = 7

Reason CONT’D: e. If 2 + 2 = 22, then x – 2 = 5. p: 2+2 = 22 q: x – 2 = 5 FALSE Refer to the truth table on slide three. Since p is false, the truth value of q is immaterial for the conclusion to be true. Hence, x may be any number.

Reason CONT’D: f. If x + 2 = 6, then 3 + 2 = 5. p: x + 2 = 6 q: 3 + 2 = 5 True Refer to the truth table on slide three. Since q is true, the truth value of p is immaterial for the conclusion to be true. Hence, x may be any number.

Definition of the Biconditional The biconditional is symbolized by p ↔ q (read as p if and only if q or p biconditional q).

Truth Value of the Biconditional Statement The biconditional statement p ↔ q is true only when p and q have the same truth values; otherwise, it is false. p q p ↔ q T F

Conditional Equivalence p q p → q T F p q ~p (~ p v q) T F T F T F

Examples Write each statement in symbolic form using ~ and ∨. Also write the statements in words. Recall that (p → q) ⇔ (~ p ∨ q). a. If the temperature is above 80˚ (a), then I will go to the beach (b). ~a ∨ b The temperature is not above 80˚, or I would go to the beach. b. If Mida is home by 5 (h), then dinner will be ready by 6 (r). ~h ∨ r Mida is not home by 5 or dinner will be ready by 6.

The statement “All even numbers are divisible by 2” can be translated as “If it is an even number, then it is divisible by 2.” In general, the statement “All ___ are ___” can be translated as “If it is a ___, then it is a ___.” Example: All dogs are mammals. If it is a dog, then it is a mammal.

Negation of a Conditional Statement p q p → q ~(p → q) T F p q ~p ~ p ∨ q ~ (~ p ∨ q) T F F T F T

Negation of a Conditional Statement CONT”D p q ~p ~ p ∨ q ~ (~ p ∨ q) T F p q ~q p Λ ~q T F F T F T

Examples Slides 13 and 14 verified that ~(p → q) is equivalent to p Λ ~q. This means that to negate an “if ___, then ___” statement, simply assert the if clause and deny the then clause. Write in words the negation of each statement. a. If you earn a lot of money, then you pay heavy taxes. You earn a lot of money but you do not pay heavy taxes. b. If Alice passes the test, then she gets the job. Alice passes the test but she does not get the job. END 17