Conditional Statements
·Conditional Statements
Can be written in the form
If A, then B.
If the popcorn burns, then the heat was too high or the kernels heated unevenly
The Hypothesis is the part of the statement immediately following "If"
The Conclusion if the part of the statement immediately following "then"
·What is the purpose of the previous sentence?
It gives two reasons why the popcorn might burn.
·Does this mean the popcorn will burn? Explain.
No: the sentence begins with "if"
·Suppose the heat is too high. What does the sentence tell you about this situation?
The popcorn will likely burn.
Identify the hypothesis and conclusion of each statement. The write each statement in "if-then" form.
·I will go to the ball game with you on Saturday
Hypothesis:
it is Saturday
Conclusion:
I will go to the ball game with you
If
it is Saturday, then I will go to the ball game with you.
·For a number x such that 6x - 8 = 16, x = 4
Hypothesis:
6x - 8 = 16
Conclusion:
x=4
If
6x - 8 = 16,
then
·Brianna wears goggles when she is swimming
Hypothesis:
Brianna is swimming
Conclusion:
she wears goggles
If
Brianna is swimming,
x=4
then
she wearing goggles
Identify the hypothesis and conclusion of each statement. Then write each statement in "if-then" form.
·Colin watches television when he does not have homework.
Hypothesis:
Colin does not have homework
Conclusion:
he watches television
If
Colin does not have homework,
then
he watches television.
·A number that is divisible by 10 is also dividible by 5
Hypothesis:
a number is divisible by 10
Conclusion:
it is divisible by 5
If
a number is divisible by 10, then
it is divisible by 5
·A rectangle is a quadrilateral with four right angles.
Hypothesis:
a quadrilateral with four right angles
Conclusion:
rectangle
If
a quadrilateral with four right angles,
then
it is a rectangle.
Deductive Reasoning
·Deductive Reasoning is the process of using facts, rules, definitions, or properties to reach a valid conclusion.
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 two numbers are odd, then their sum is even."
·
The two numbers are 7 and 3
7 and 3 are odd, so the jypothesis is true.
Conclusion: the sum of 7 and 3 is even
·The sum of two numbers is 14
The conclusion is true. If the numbers are 11 and 3, the
hypothesis is true also. However, if the numbers are 8 and 6, the
hypothesis is false. Therefore, there is no valid conclusion that
can be drawn from the given conditional.
Determine a valid conclusion that follows from the statement below for each given condition. If a valid
conclusion does not follow, write "no valid conslusion" and explain why.
If the last digit of a number is 2, then the number is divisble by 2.
·The number is 10, 452.
The number is divisible by 2.
·The number is divisible by 2.
No valid conclusion: the last digit could be any
even number.
·The number is 946.
No valid conclusion; the last digit is 6.
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.
5 is odd and 12 is even, and 5 + 12 = 17.
Conclusion:
The sum of 5 and 12 is odd.
·The two numbers are 8 and 26.
8 and 26 are both even so the hypothesis is false. No valid
conclusion.
Determine a valid conclusion that follows from the statement below for each condition.
"There will be a quiz every Wednesday."
It is Wenesday
It is Tuesday
There will be a quiz.
There will might not be a quiz today.
Counterexamples
A Counterexample is a specific case in which the hypothesis is true and the conclusion is false. It only takes
one counterexample to show that a statement is false.
·"If a triangle has a perimeter of 3 cm, then each side measures 1 cm."
Counterexample: The sides are 0.9cm, 0.9cm, and 1.2 cm.
·Which numbers are counterexamples to the following statement?
x - y ≠y - x
A)
x = 2, y = 3
B)
x = 4, y = 4
C)
x = 0, y = 1
D)
x = 7, y = -7
B
·Which numbers disprove the statement below?
If x + y > xy, then x>y
F)
x = 1, y = 2
G)
x = 2, y = 3
H)
x = 4, y = 1
J)
x = 4, y = 2
F
Find a counterexample for each conditional statement
·If Anna is in school, then she has a science class.
Anna has a schedule without a science class.
·If you can read 8 pages in 30 minutes, then you can read a book in a day.
A book can have more than 384 pages
·If a number x is squared, then x2 > x
·If 3x + 7 ≥ 52, then x > 15
x=1
x = 15