Lesson 2.2 Analyze Conditional Statements

Goal: The learner will write definitions as conditional statements.

Vocabulary

Conditional Statement: a logical statement that has a hypothesis and a conclusion. If-then form: format for a conditional

statement. Hypothesis: the “if” part Conclusion: the “then” part Hypothesis must always be true.

Example

If it is raining, then there are clouds in the sky.

If the car is a Mustang, then it is a Ford.

Writing Conditional Statements

All birds have feathers.

Two angles are supplementary if they are a linear pair.

More Examples

All 90° angles are right angles.

When n = 9, n² = 81.

Tourists at the Alamo are in Texas.

Negation: the opposite of the original statement.

The ball is red. Negation:

The cat is not black. Negation:

Converse: flip-flop the hypothesis and conclusion.

If it is raining, then I will carry an umbrella. Converse

If I am in Roadtown, then I’m in Tortola. Converse

Inverse: Negate both the hypothesis and conclusion

If it is a Corvette, then it is a Chevy. Inverse:

If you are a soccer player, then you are an athlete. Inverse:

Contrapositive: negate and flip-flop the hypothesis and conclusion.

obtuse. isA then ,99A If Converse:

Inverse:

Contrapositive:

Are these statements true?

Write the if-then, converse, inverse, and contrapositive for the conditional statement: All whales are mammals

If-then: If it is a whale, then it is a mammal.

Converse: If it is a mammal, then it is a whale.

Inverse: If it is not a whale, then it is not a mammal.

Contrapositive: If it is not a mammal, then it is not a whale.

Verifying Statements

You must show the conclusion is true every time the hypothesis is true.

It only takes one counterexample to show it’s false.

Use “Guitar players are musicians.” to write the following.

“If-then”

Converse

Inverse

Contrapositive

Which statements are true? Give a counterexample if it is false.

If a polygon is equilateral, then the polygon is regular. Converse

Inverse

Contrapositive

Equivalent Statements: when two statements are both true or false.

Conditional Statement and its contrapositive are either both true or false.

Converse and inverse are either both true or false.

Definitions as Conditional Statments

Any definition can be written as “if-then” or as its converse.

Example: Right Angles: If the angle measure is

90◦, then it is a right angle.

Biconditional Statement: When a “If-Then” and its converse are true you can write them as a single statement.

All definitions are biconditional. Example:

Perpendicular lines: If the angle measure is 90◦, then then it is a right angle.

Converse: If the angle is a right angle, then the its measure is 90◦.

Biconditional: An angle is a right angle if and only if the its measure is 90◦

Example

Write the definition of straight angle as a biconditional statement.

Another Example

If Mary is in theater class, she will be in the fall play. If Mary is in the fall play, she must be taking theater class.