2.2 Conditional Statements

Conditional Statement

A statement of the form 'If p, then q' where p is the hypothesis and q is the conclusion.

Hypothesis

The 'if' part of a conditional statement, denoted by p.

Conclusion

The 'then' part of a conditional statement, denoted by q.

Truth Value

The truth value of a conditional statement is false only when p is true and q is false; true otherwise.

Vacuously True

A conditional statement is considered vacuously true if its hypothesis is false, regardless of the conclusion.

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.

Biconditional Statement

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

Sufficient Condition

A condition r is sufficient for s means 'if r then s'.

Necessary Condition

A condition s is necessary for r means 'if not r then not s'.

Logical Equivalence

Two statements are logically equivalent if they have the same truth value in all scenarios.

Order of Operations (Conditionals)

In complex logical expressions, the conditional operator (→) is evaluated last.

Negation of a Conditional Statement

The negation of a conditional statement ~(p → q) is logically equivalent to p ∧ ~q.

Truth Table for Conditional Statements

A table that shows the truth values of p, q, and p → q.

Example of Conditional Statement

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

Only If

"p only if q" means "if not q then not p," or equivalently, "if p then q."

Hierarchy of Operations for Logical Operators

1. ~ (Negation) 2. ∧ (Conjunction), ∨ (Disjunction) 3. → (Conditional), ↔ (Biconditional)

Division into Cases

A technique using truth tables to prove logical equivalence by dividing the analysis into cases based on the truth values of the variables.

Representing Conditional Statements Using Only ~, ^, or v

A method to express p → q using only negation (~), conjunction (^), and/or disjunction (v), typically resulting in an expression like ~(p ∧ ~q) or ~p ∨ q.

Conditional Statements with False Hypothesis

Even if the hypothesis is false, the conditional statement is considered true. This is often called "vacuously true" or "true by default."

Relationship Between Converse and Inverse

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

Converting "Only If" to "If-Then"

A statement "p only if q" can be converted to "If p, then q." The contrapositive of this is "If not q, then not p."

Biconditional as Conjunction of Two If-Then Statements

A biconditional statement "p if and only if q" can be expressed as the conjunction of two if-then statements: "If p, then q" and "If q, then p."