1-3_Implications_and_Biconditionals

1.3 Implications and Biconditionals

Overview

• Importance of Implications: Statements of the form "If P, then Q" are fundamental in mathematics.

  • Example from Precalculus: "If two lines in a plane have the same slope, then the lines are parallel."

  • Example from Calculus: "If a function f is differentiable at x = c, then f is continuous at x = c."

Implications

• Definition 1.3.1: An implication (conditional) is a statement of the form "If P, then Q" denoted by P ⇒ Q.

  • Terminology:

    • P: Assumption (hypothesis)

    • Q: Conclusion (consequent)

• Example 1.3.2: Statement: "If an integer p is a prime that divides ab, then p divides a or b."

  • Form: (P ∧ Q) ⇒ (R ∨ S) where:

    • P: "p is a prime"

    • Q: "p divides ab"

    • R: "p divides a"

    • S: "p divides b"

Truth Values of Implications

• Remark 1.3.3: The implication P ⇒ Q is false only when P is true and Q is false.

  • Truth table for P ⇒ Q:

    • P Q | P ⇒ Q

    • T T | T

    • T F | F

    • F T | T

    • F F | T

• Example Explanation:

  • Scenario: "If you earn an A on the final exam, then you will receive an A for your final grade."

  • If P (earning an A) is false, the implication is always considered true.

  • No promises made if the assumption is false.

Example with Truth Values

• Example 1.3.4: Consider statements P: "17 is even" and Q: "19 is prime."

  • (1) P ⇒ Q: "If 17 is even, then 19 is prime" - True (P is false)

  • (2) Q ⇒ P: "If 19 is prime, then 17 is even" - False (Q is true, P is false)

  • (3) (¬P) ⇒ (¬Q): "If 17 is not even, then 19 is not prime" - False (¬P true, ¬Q false)

  • (4) (¬Q) ⇒ (¬P): "If 19 is not prime, then 17 is not even" - True (¬Q false)

Converse and Contrapositive

• Definition 1.3.6:

  • The converse of P ⇒ Q is Q ⇒ P.

  • The contrapositive of P ⇒ Q is (¬Q) ⇒ (¬P).

• Example 1.3.7:

  • Implication: "If π is an integer, then 14 is even."

    • Converse: "If 14 is even, then π is an integer."

    • Contrapositive: "If 14 is not even, then π is not an integer."

Biconditional Statements

• Definition 1.3.8: A biconditional statement is the conjunction of both implications: (P ⇒ Q) ∧ (Q ⇒ P).

  • Denoted as P ↔ Q and read as "P if and only if Q."

Truth Values of Biconditional Statements

• Remark 1.3.9: The truth table for P ↔ Q is:

  • P Q | P ⇒ Q | Q ⇒ P | P ↔ Q

  • T T | T | T | T

  • T F | F | T | F

  • F T | T | F | F

  • F F | T | T | T

• Definition Criteria: Definitions using "if and only if" are key examples of biconditionals.

• Example 1.3.10: Truth value determinations:

  • (1) "7 is an odd integer if and only if 14 is prime" - False

  • (2) "102 is an even integer if and only if 23 is prime" - True

  • (3) "3 is even if and only if 9 is prime" - True

Exercises

• Exercise 1.3.11: Truth Table Construction

  • For (¬P) ⇒ Q ∨ (Q, P).

• Exercise 1.3.12: Truth Table Construction for (¬Q) ⇒ (Q, P).