2.2 Conditional Statements

Goal: Students will be able: To recognize conditional statements and their parts. To write converses, inverses, and contrapositives

of conditional statements.

Conditional Statement A statement that can be written in “if – then”

form. Symbol: p → q, read if p then q, or p implies q.

Example:

If it rains on Thursday, then the baseball game will be canceled.

pp

q

Hypothesis The phrase immediately following the word if in a

conditional statement The p part following if.

The phrase immediately following the word then in a conditional statement.

Conclusion

If p, then q.

Hypothesis Conclusion

Examples 1: Determine Hypothesis and conclusion

If trout are fish, then trout live in a pond.

If you buy a car, then you get $1500 cash back.

Hypothesis: trout are fishConclusion: trout live in a pond

Hypothesis: you buy a carConclusion: you get $1500 cash back

Got it 1? Identify the hypothesis and conclusion of each statement.

If an animal is a robin, then the animal is a bird.

If an angle measures 180°, then the angle is obtuse.

If a polygon has 6 sides, then it is a hexagon.

Hypothesis: an animal is a robinConclusion: the animal is a bird.

Hypothesis: an angle measures 180°Conclusion: the angle is obtuse

Hypothesis: a polygon has 6 sidesConclusion: it is a hexagon

Example 2: Writing a Conditional

How can you write the following statement as a conditional?

Vertical angles share a vertex.

Step 1: Identify the hypothesis and conclusion.

In order for two angles to be vertical, they must share a vertex.

So the set of vertical angles is inside the set of angles that share a vertex.

Hypothesis: Vertical angles Conclusion: share a vertex.

If two angles are vertical angles, then they share a vertex.

Got it 2? How can you write “Dolphins are mammals” as a conditional?

Mammals

Dolphins

If an animal is a dolphin, then it is a mammal.

Truth Value Is either 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, find only one

counterexample, where the hypothesis is true, and the conclusion is false.

Example 3: Finding the truth value of a conditional.

Is the conditional true or false? If it is false, find a counterexample.

If a number is divisible by 3, then it is odd.

The conclusion is false. The number 12 is divisible by 3, and 12 is even.

Got it? 3: Finding the truth value of a conditional.

Is the conditional true or false? If it is false, find a counterexample.

If a month has 28 days, then it is February.

If two angles form a linear pair, then they are supplementary.

False, January has 28 days plus 3 more.

True

Negation Is the opposite of the original statement

~p: The sky is not blue.

p: The sky is blue.Examples:

~p, read not p Symbol: ~

~q: A triangle does not have 4 sides.

q: A triangle has 4 sides.

Converse, Inverse, and ContrapositiveStatement Formed by Symbols Examples

Conditional

Converse

Inverse

Contrapositive

Given hypothesisand conclusion

p → qIf two angles have the samemeasure, then they are congruent.

Exchange the hypothesis and conclusion of the conditional

q → pIf two angles are congruent,then they have the sameMeasure.

Negate both the hypothesis and conclusion of the conditional

~p → ~qIf two angles do not havethe same measure, thenthey are not congruent.

Exchange and Negating both the hypothesis and conclusion of theconditional

~q → ~pIf two angles are notcongruent, then they do nothave the same measure.

Equivalent Statements Have the same truth value

The conditional and the contrapositive are equivalent statements.

The converse and the inverse are equivalent statements.

Biconditional Statements: When a conditional statement and its converse are

both true, you can write them as a single biconditional statement.

A biconditional statement is a statement that contains the phrase “if and only if.”

Any valid definition can be written as a biconditional statement.

Example 1: Rewrite the conditional statement in if-then form.

All birds have feathers.

Conditional: If it is a bird, then it has feathers.

Conditional: If two angles are a linear pair, then they are supplementary.

Two angles are supplementary if they are a linear pair.

Example 2: Write the converse, inverse, and contrapositive of the following conditional?

If a dog is a Great Dane, then it is large.

Converse: If the dog is large, then it is a Great Dane.

Inverse: If the dog is not a Great Dane, then it is not large.

Contrapositive: If the dog is not large, then it is not a Great Dane.

Example 4. Write the definition of perpendicular lines as a biconditional.

Definition: If two lines intersect to form a right angle, then they are perpendicular.

Converse: If two lines are perpendicular, then they intersect to form right angles.

Biconditional: Lines intersect to form right angles iff they are perpendicular lines.