17 Logical Reasoning

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1-7 Logical Reasoning
Algebra 1 Glencoe McGraw-Hill Linda
Stamper
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Logical Reasoning includes conditional
statements, deductive reasoning, and
counterexamples.
A conditional statement has a hypothesis and a
conclusion and is often written in if-then form.
Deductive reasoning is a process that uses facts
and rules to reach a valid conclusion.
A counterexample is a specific example that can
be used to show that a statement is false.
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Example of a conditional statement
If the popcorn burns, then the heat was too high.
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The part of the statement immediately following
if is called the hypothesis.
The part of the statement immediately following
then is called the conclusion.
Identify the hypothesis and conclusion of the
statement.
If it is Friday, then the Smiths are going out to
dinner.
hypothesis it is Friday
conclusion the Smiths are going out to dinner
Note that then is not part of the conclusion.
Note that if is not part of the hypotheses.
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Identify the hypothesis and conclusion of each
statement.
Example 1 If it is raining, then the party will
be indoors.
hypothesis it is raining conclusion the party
will be indoors
Example 2 If 4x 3 gt 27, then x gt 6.
hypothesis 4x 3 gt 27 conclusion x gt 6
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Deductive reasoning is the process of using
facts, rules, definitions, or properties to reach
a valid conclusion.
You can use deductive reasoning to determine
whether a valid conclusion follows from a
conditional statement.
Determine a valid conclusion that follows from
the conditional statement below. Explain your
answer.
Conditional statement If two numbers are odd,
then their sum is even.
Given condition The two numbers are 7 and 3.
7 and 3 are odd so the hypotheses is true. The
sum of 7 and 3 is even so the conclusion is valid.
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Example 3 Determine a valid conclusion that
follows from the conditional statement below.
Conditional statement There will be a quiz
every Wednesday.
Given condition It is Wednesday.
Valid conclusion _______________________
There will be a quiz.
Example 4 Determine a valid conclusion that
follows from the conditional statement below.
Conditional statement If your test score is in
the 90th percentile, then your grade is an A.
Given condition score is 95
Valid conclusion ________________________
Your grade is an A.
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To show that a condition is false, we can use a
counterexample. A counterexample is a specific
case in which the hypotheses is true and the
conclusion is false.
Conditional statement If a triangle has a
perimeter of 3 inches, then each side measure is
1 inch.
A counterexample is a triangle with perimeter of
3 and sides of 0.9, 0.9, and 1.2.
It takes only one counterexample to show that a
statement is false.
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Example 5 Find a counterexample to show that the
conditional statement is false.
Conditional statement If you graduate from
Colina, then you go to Westlake High.
Counterexample _______________________
A graduate attends TOHS.
Example 6 Find a counterexample to show that the
conditional statement is false.
Conditional statement If x 3 gt -6, then x
must be negative.
Counterexample ___________________________
When x is 10 the statement is true
and 10 is a positive number.
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Example 7 Find a counterexample to show that the
conditional statement is false.
Conditional statement Every four-sided figure
is a rectangle.
Counterexample
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Homework

• 1-A12 Pages 42-44, 15-24, 29-34, 47-49
• Page 57, 10-18