Section 1'1 Inductive and Deductive Reasoning

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Section 1.1 Inductive and Deductive Reasoning

• Thinking Mathematically
• Math 1001

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Section 1.1 Objectives

• Understand and use inductive reasoning.
• Understand and use deductive reasoning.

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What is the next number?

• 2, 4, 8, 16,
• 32
• 3, 7, 11, 15,
• 19

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Inductive Reasoning (page 3)

• Inductive reasoning is the process of arriving at
  a general conclusion based on observations of
  specific examples.
• Inductive reasoning does not prove that the
  pattern applies in all cases.
• The conclusion is called a conjecture,
  hypothesis, or an educated guess.

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Inductive Reasoning
Inductive reasoning is generalizing from
experience to reach a conclusion.
Conclusion College textbooks cost more than
50.00.
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Inductive Reasoning
There is no guarantee that the conclusions
reached by inductive reasoning are correct with
no exceptions.
Even though all your textbooks this term cost
more than 50.00 it is still possible (and even
probable) that a textbook for some course will
cost less than 50.00.
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Inductive Reasoning
In mathematics inductive reasoning is often
used to find patterns.
Find the pattern in the following sequence of 6
numbers and use that pattern to decide what the
next number should be.
1,4,16,64,256,1024,?
Each number is obtained from the previous one by
multiplying by 4. The next number is
4106
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Conjecture and Counterexample

• If there is just one case where the conjecture
  does not work then the conjecture is false.
• Such a case is called a counterexample.

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Example 2

• Conjecture The sum of any two two-digit numbers
  is always a three-digit number.

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Check Point 2

• Conjecture the product of two two-digit numbers
  is a three digit number.

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Deductive Reasoning (page 6)

• Deductive reasoning is the process of proving a
  specific conclusion from one or more general
  statements.
• A conclusion that is proved true by deductive
  reasoning is called a theorem.

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Deductive Reasoning
Deductive reasoning arrives at conclusions from
general statements that are assumed to be true.
You are an entering freshman.
Conclusion You will have to take a mathematics
placement test.
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Example 5

• Select a number
• Multiply by 6
• Add 8 to the product
• Divide the sum by 2
• Subtract 4 from the quotient

• 4, 7, 11, 100
• 12
• 21
• 33
• 300

Make a conjecture
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Example 5

• Select a number
• Multiply by 6
• Add 8 to the product
• Divide the sum by 2
• Subtract 4 from the quotient

Proof of the conjecture
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Section 1.1 Homework

• Pg 8-10,
• 1-32, 35-36, 38-39, 42-44,47-48