2.2 Conditional Statements

Definition and Structure

Conditional Statement

A statement of the form "If p, then q" where p is the hypothesis and q is the conclusion.Notation: Denoted as p → q.Truth Value: False only when p is true and q is false; true otherwise.Hypothesis (or antecedent): The "if" part of the statement, denoted by p.Conclusion (or consequent): The "then" part of the statement, denoted by q.

Example

"If 4,686 is divisible by 6, then 4,686 is divisible by 3."

• Hypothesis (p): 4,686 is divisible by 6.

• Conclusion (q): 4,686 is divisible by 3.

Truth Table for Conditional Statements

Vacuously True

A conditional statement is considered true if its hypothesis is false, regardless of the truth value of the conclusion.Example: "If the moon is made of cheese, then I am a millionaire."This statement is vacuously true because the hypothesis ("the moon is made of cheese") is false.

Understanding Conditional Statements

Promise Analogy

A conditional statement can be thought of as a promise. It is only broken (false) if the promise is made (p is true) and not kept (q is false).

False Hypothesis

If the hypothesis is false, the conditional statement is always true, regardless of the truth value of the conclusion.

Example

"If you get at least 90 on your exam, then you will get grade A."

• Case 1: You got 90, and you got A. (Promise kept)

• Case 2: You got 90, and you did not get A. (Promise broken)

• Case 3: You did not get 90, and you got A. (Promise not broken)

• Case 4: You did not get 90, and you did not get A. (Promise not broken)

Logical Equivalences Involving Conditional Statements

Order of Operations

In complex logical expressions, the conditional operator (→) is evaluated last. Representation of If-Then as Or: p → q is logically equivalent to ~p ∨ q.

Negation of a Conditional Statement: ~(p → q) is logically equivalent to p ∧ ~q.

Example

"If it is raining, then the ground is wet."

• Equivalent or statement: "It is not raining or the ground is wet."

• Negation: "It is raining and the ground is not wet."

Related Forms of Conditional Statements

Contrapositive

The contrapositive of p → q is ~q → ~p. A conditional statement and its contrapositive are logically equivalent.

Converse

The converse of p → q is q → p.

Inverse

The inverse of p → q is ~p → ~q.

Relationship

The converse and inverse of a conditional statement are logically equivalent to each other, but neither is equivalent to the original conditional.

Example

"If it is a cat, then it is a mammal."

• Contrapositive: "If it is not a mammal, then it is not a cat."

• Converse: "If it is a mammal, then it is a cat."

• Inverse: "If it is not a cat, then it is not a mammal."

Biconditional Statements

Definition

A statement of the form "p if, and only if, q" denoted as p ↔ q.

Truth Value

True if p and q have the same truth value, false otherwise.

Truth Table

Order of Operations

The biconditional operator (↔) has the same precedence as the conditional operator (→). Parentheses are needed to indicate precedence between them.

Example

"A triangle is equilateral if, and only if, it is equiangular."

Necessary and Sufficient Conditions

Sufficient Condition

r is a sufficient condition for s means "if r then s".

Necessary Condition

s is a necessary condition for r means "if not r then not s" or equivalently "if s then r".

Example

"Being a square is a sufficient condition for being a rectangle." (If a shape is a square, then it is a rectangle.)"Being a rectangle is a necessary condition for being a square." (If a shape is not a rectangle, then it is not a square.)

This guide provides a comprehensive overview of the key concepts related to conditional statements, their logical equivalences, and related forms. Understanding these concepts is crucial for mastering logical reasoning and problem-solving in various fields.