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1.7 Logical Reasoning • Conditional statements – written in the form If A, then B. • Statements in this form are called if-then statements. – Ex. If the popcorn burns, then the heat was too high or the kernels heated unevenly. • The part of the statement immediately following the word if is called the hypothesis. • The part of the statement immediately following the word then is called the conclusion.

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Page 1: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

1.7 Logical Reasoning• Conditional statements – written in the form If A,

then B.• Statements in this form are called if-then

statements.– Ex. If the popcorn burns, then the heat was too high or

the kernels heated unevenly.• The part of the statement immediately following the word if is

called the hypothesis.• The part of the statement immediately following the word

then is called the conclusion.

Page 2: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Identify Hypothesis and Conclusion• Identify the hypothesis and conclusion of each

statement.a. If it is Friday, then Madison and Miguel are going to

the movies.Hypothesis: it is FridayConclusion: Madison and Miguel are going to the movies.

b.If 4x + 3 > 27, then x > 6.Hypothesis: 4x + 3 > 27Conclusion: x > 6

Page 3: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Conditionals

• Sometimes a conditional statement is written without using the words if and then.

• A conditional statement can always be rewritten as an if-then statement.– Ex. When it is not raining, I ride my bike.• Rewritten: If it is not raining, then I ride my bike.

Page 4: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Write a Conditional in If-Then Form• Identify the hypothesis and conclusion of each

statement. Then write each statement in if-then form.

a.I will go to the ball game with you on Saturday.Hypothesis: It is SaturdayConclusion: I will go to the ball game with youIf it is Saturday, then I will go to the ball game with you.

Page 5: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Write a Conditional in If-Then Form• Identify the hypothesis and conclusion of each

statement. Then write each statement in if-then form.

b. For a number x such that 6x – 8 = 16, then x = 4.Hypothesis: 6x – 8 = 16Conclusion: x = 4If 6x – 8 = 16, then x = 4.

Page 6: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Deductive Reasoning and Counterexamples• 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.

Page 7: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Deductive Reasoning• Determine a valid conclusion that follows from the statement

“If two numbers are odd, then their sum is even” for the given conditions. If a valid conclusion does not follow, write no valid conclusion and explain why.

a. The two numbers are 7 and 3.7 and 3 are odd, so the hypothesis is true.Conclusion: The sum of 7 and 3 is even.CHECK: 7 + 3 = 10 The sum, 10, is even.

b. 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. There is no way to determine the two numbers. Therefore, there is no valid conclusion.

Page 8: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Counterexample

• To show that a conditional is false, we can use a counterexample.

• A counterexample is a specific case in which a statement is false.

• It takes only one counterexample to show that a statement is false.

Page 9: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Find Counterexample• Find a counterexample for each conditional

statement.a.If you are using the Internet, then you own a

computer.You could use the Internet on a computer at a library.

b.If the Commutative Property holds for multiplication, then it holds for division. 2 ÷ 1 ≠ 1 ÷ 2

Page 10: 1.7 Logical Reasoning Conditional statements – written in the form If A, then B. Statements in this form are called if-then statements. – Ex. If the popcorn

Find a Counterexample• Which numbers are counterexamples for the

statement below?If x ÷ y = 1, then x and y are whole numbers.

a. x = 2, y = 2 b. x = 0.25, y = 0.25c. x = 1.2, y = 0.6 d. x = 6, y = 3

The only values that prove the statement false are x = 0.25 and y = 0.25. So, these numbers are counterexamples. The answer is B.


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