2.2 conditional statements goal: students will be able: to recognize conditional statements and...
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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.
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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
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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
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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
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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
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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.
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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.
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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.
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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.
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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
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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.
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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.
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Equivalent Statements Have the same truth value
The conditional and the contrapositive are equivalent statements.
The converse and the inverse are equivalent statements.
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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.
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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.
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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.
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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.