In Chapter 2, Section 3, we learned about Conditional Statements.

A conditional statement is a statement that can be written in if-then form. When a statement is in if-then form, it is written in the form if p, then q. The part of the sentence after the word "if" is the hypothesis, and the part of the sentence after the word "then" is the conclusion.

Example 1: If you buy the television, then you will get $100 cash back.
hypothesis- You buy the television
conclusion- You get $100 cash back

Example 2: She will get a promotion if she works hard.
hypothesis- She will get a promotion
conclusion- She works hard.
hypothesis- She will get a promotion.

*There is no if-then statement in example 2. However, you can identify the hypothesis and conclusion by first identifying the hypothesis and then finding the conclusion.

Writing a Conditional Statement

First, identify the hypothesis and conclusion. Then, put them into a statement using if-then form.

Examples:

The boy is Tom's son, and he is four feet tall.
hypothesis- The boy is Tom's son
conclusion- He is four feet tall.
Conditional Statement: If he the boy is Tom's son, then he is four feet tall.

An angle with a measure of 180 degrees is a straight angle.
hypothesis- An angle has a measure of 180 degrees.
conclusion- it is a straight angle.
Conditional Statement: If an angle has a measure of 180 degrees, then is it a straight angle.

Finding the Truth Value of a Statement

The truth value of a statement is either true or false. Both parts of the statement, the hypothesis and the conclusion, must be true in order for the statement to be true.

Example:

If there is a sale, then you will buy her a pair of shoes.

a. There was a sale at the mall; you bought her a pair of shoes.
The hypothesis is true, because there was in fact a sale. The conclusion is true, because you bought her a pair of shoes, which is what was promised in the statement. Both parts are true, and therefore, the conditional statement is TRUE.

b. There was a sale at the mall; you bought her two pairs of shoes.
The hypothesis is true, because there was a sale. The conclusion is false, because you bought her two pairs of shoes, which is not what was promised. Both parts are NOT true, and therefore, the conditional statement is FALSE.

Related Conditionals

Statements based on a conditional statement are called related conditionals.

Here is a chart of the related conditonals:


Example:
Conditional: If you eat healthy food and exercise daily, then you are a healthy person.
Converse: If you are a healthy person, then you eat healthy food and exercise daily.
Inverse: If you do not eat healthy food and do not exercise daily, then you are not a healthy person.
Contrapositive: If you are not a healthy person, then do do not eat healthy food and exercise daily.

Statements with the same value are considered logically equivalent.