Section 2-1

Conditional Statements

 

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What You'll Learn

  1. To recognize conditional statements
  2. To write converses of conditional statements

… And Why

To help you read critically, as in Example 7

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Conditional Statements

You have heard if-then statements such as this one:

If you are not completely satisfied, then your money will be refunded.

Another name for an if-then statement is a conditional. Every conditional has two parts. The part following if is the hypothesis, and the part following then is the conclusion.

Example 1 Identifying the Hypothesis and the Conclusion

Identify the hypothesis and the conclusion of this conditional statement:

If today is the first day of fall, then the month is September.

Hypothesis: Today is the first day of fall.
Conclusion: The month is September.


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You can write many sentences as conditionals.

Example 2 Writing a Conditional

Write each sentence as a conditional.

  1. A rectangle has four right angles.
    If a figure is a rectangle, then it has four right angles.
  2. A tiger is an animal.
    If something is a tiger, then it is an animal.

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A conditional can have a truth value of true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.

Image of 2 calendars for the month of February, for years 2007 and 2008. February 2007 has 28 days and February 2008 has 29 days. 

Example 3 Finding a Counterexample

Show that this conditional is false by finding a counterexample:

If it is February, then there are only 28 days in the month.

To show that this conditional is false, you need to find one counterexample that makes the hypothesis true and the conclusion false.

February in the year 2008 is a counterexample. Because 2008 is a leap year, the month of February has 29 days.

The conditional is false because February 2008 is a counterexample.


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You can use a Venn diagram to better understand true conditional statements.

Example 4 Using a Venn Diagram

Draw a Venn diagram to illustrate this conditional:

If you live in Chicago, then you live in Illinois.

The set of things that satisfy the hypothesis lies inside the set of things that satisfy the conclusion.


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