alg 1 1­7.notebook
1‐7, Logical Reasoning and Counterexamples
The statement If the popcorn burns, then the heat was too
high or the kernels heated unevenly is called a conditional
statement. Conditional statements can be written in the form
If A, then B. Statements in this form are called if‐then
statements.
August 29, 2012
If the popcorn burns, then the heat was too high or the
kernels heated unevenly
What is the purpose of this sentence?
Does this mean the popcorn will burn?
Suppose the heat is too high. What does the sentence tell you
about this situation?
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 the conclusion of each statement.
If it is warm this afternoon, then we will have the party
outside.
If 8w ‐ 5 = 11, then w = 2.
Sometimes a conditional statement is written without using
the words if and then. But a conditional statement can always
be rewritten in if‐then form.
Identify the hypothesis and conclusion of each statement.
Then write each statement in if‐then form.
Brianna wears goggles when she is swimming.
A rhombus with side lengths of (x ‐ y) units has a perimeter of
(4x ‐ 4y) units.
alg 1 1­7.notebook
Deductive reasoning is the process of using facts, rules,
definitions, or properties to reach a valid conclusion. Suppose
you have a true conditional and you know that the hypothesis
is true for a given case. Deductive reasoning allows you to say
that the conclusion is true for that case.
August 29, 2012
Determine a valid conclusion that follows from the statement
"There will be a quiz every Wednesday."
It is Wednesday.
It is Tuesday.
To show that a conditional is false, we can use a
counterexample. A counterexample is a specific case in which
the hypothesis is true and the conclusion is false. For
example, consider the conditional if a triangle has a
perimeter of 3 centimeters, then each side measures 1
centimeter. A counterexample is a triangle with perimeter 3
and sides 0.9, 0.9, and 1.2 centimeters long. It takes only one
counterexample to show that a statement is false.
Determine a valid conclusion that follows from the statement
below for each condition. If a valid conclusion does not follow,
write no valid conclusion and explain why.
If one number is odd and another number is even, then
their sum is odd.
The two numbers are 5 and 12.
The two numbers are 8 and 26.
Identify the hypothesis and conclusion of each statement.
Identify the hypothesis and conclusion of each statement.
Then write each statement in if‐then form.
If it is April, then it might rain.
Colin watches television when he does not have homework.
If Colin watches television, then he does not have homework.
If you play tennis, then you run fast.
A number that is divisible by 10 is also divisible by 5.
If a number is divisible by 10, then it is also divisible by 5.
If 34 ‐ 3x = 16, then x = 6.
A rectangle is a quadrilateral with four right angles.
If a quadrilateral is a rectangle, then it has four right angles.
alg 1 1­7.notebook
Determine a valid conclusion that follows from the statement
below for each given condition. If a valid conclusion does not
follow, write no valid conclusion and explain why.
If the last digit of a number is 2, then the number is
divisible by 2.
August 29, 2012
Find a counterexample for each conditional statement.
If Anna is in school, then she has a science class.
If you can read 8 pages in 30 minutes, then you can read a
book in a day.
The number is 10,452.
If a number x is squared, then x 2 > x.
The number is divisible by 2.
If 3x + 7 ≥ 52, then x > 15.
The number is 946.