Conditional Statements - Section 1.5

A set of QUESTION_AND_ANSWER flashcards covering conditional statements, including definitions, relationships (converse, inverse, contrapositive), and biconditionals, as presented in the notes.

What is a conditional statement?

An if-then statement relating a hypothesis p to a conclusion q, written as p → q.

In a conditional, what do p and q represent?

p is the hypothesis (the 'if' part); q is the conclusion (the 'then' part).

How is the conditional p → q read?

As 'If p, then q.'

What is the Converse of a conditional p → q?

If q, then p.

What is the Inverse of a conditional p → q?

If not p, then not q.

What is the Contrapositive of a conditional p → q?

If not q, then not p.

What is a Biconditional?

A statement that combines a conditional and its converse: p if and only if q (p ↔ q).

When is a biconditional true?

When p and q have the same truth value (both true or both false).

What symbol denotes a biconditional?

p if and only if q, written as p ↔ q.

What is a truth value?

The truth value of a statement is whether it is true or false (T or F).

What is a truth table used for?

To list all possible truth values for two or more statements.

What determines the truth value of a conditional?

The truth value of p → q depends on the truth values of p and q; truth tables show all possibilities.

Provide an example of a conditional statement from the notes.

You can register to vote if you are at least 18 years old.

What is the Converse of 'If you play the trumpet, then you play a brass instrument'?

If you play a brass instrument, then you play the trumpet.

What are the two conditionals implied by 'A triangle is equilateral if and only if it has three congruent sides'?

If a triangle is equilateral, then it has three congruent sides; If a triangle has three congruent sides, then it is equilateral.

What are the two conditionals implied by 'The product of two numbers is negative if and only if the numbers have opposite signs'?

If the product is negative, then the numbers have opposite signs; If the numbers have opposite signs, then the product is negative.

How does a conditional differ from a biconditional?

A conditional (p → q) is one-way; a biconditional (p ↔ q) requires both directions and expresses equivalence.

What is the contrapositive of p → q?

If not q, then not p.

What is the symbol for 'If and only if'?

↔ (the biconditional symbol).

When is a biconditional false?

When p and q have opposite truth values.