Conditional Statements

Conditionals

Definition

A conditional is an if-then statement.

Symbolic

p q

read

If p, then qor

p implies q

Diagram

p

q

Rewrite the statement in if-then form. Then draw a venn diagram.

Students who study do well on tests.

x = 4 implies x2 = 16.

In an if-then statement, the if clause is the hypothesis, and the then clause is the conclusion.

If an animal is a robin, then

the animal is a bird.

Identify the hypothesis and conclusion.

The truth value of a conditional statement is either true or false.

To show a conditional true, show every time the hypothesis is true the conclusion is also true.

To show a conditional false, find ONE counterexample.

If two numbers are odd, then their sum is odd.

a. Underline the hypothesis of the statement.

b. Circle the conclusion of the statement.

c. The conditional is false. Give a counterexample for the conditional statement.

The company that prints the bumper sticker at the left below accidentally reworded the original statement and printed the sticker three different ways. Suppose the original bumper sticker is true. Are the other bumper stickers true or false? Explain.

Every conditional statement has three related conditionals.

Conditional: If p, then q. (p q)

Converse: If q, then p. (q p)

Inverse: If not p, then not q. (~p ~q)

Contrapositive: If not q, then not p. (~q ~p)

Given the conditional statement:If a figure is a triangle, then it is a polygon.

Complete the table.

Statements with the same truth values are logically equivalent.

Conditional Contrapositive

Converse Inverse

Write a true conditional statement whose inverse is false.

Write a true conditional statement that is logically equivalent to its converse.

Get in your groups….

When both a statement and its converse are true, you can connect the hypothesis and conclusion with the words “if and only if.” This is called a bi-conditional.

Determine whether the conditional statement can be written as a bi-conditional.

1) If an animal is a fish, then it swims.2) If a coin is a penny, then it is worth 1 cent.3) If two angles are supplementary, then their

sum is 180º.

Write the definition of perpendicular lines in if and only if form.

All definitions you have learned can be written as “if and only if” statements.