Download Section 1.1 Inductive and Deductive Reasoning

Section 1.1
Inductive and Deductive
Reasoning
1.
2.
Inductive Reasoning
Objectives:
Understand and use inductive
reasoning.
Understand and use deductive
reasoning.
The process of arriving at a general
conclusion based on observations of
specific examples.
Definitions:
Conjecture/hypothesis: The conclusion formed
as a result of inductive reasoning which may or
may not be true.
Counterexample: A case for which the
conjecture is not true which proves the
conjecture is false.
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Strong Inductive Argument
2
Weak Inductive Argument
This cooler contains 30 soda cans
I looked at 27 of the cans, and they were all
Coke
Probably all 30 of the cans are Coke
Conjecture?
all 30 sodas are Coke
Counterexample?
if you pull out a soda that is not Coke
Since neither my dad nor my brother ever
cried in front of me, men have difficulty
expressing their feelings.
This conclusion is based on just two
observations.
This sample is neither random nor large
enough to represent all men.
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Example 1
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Example 1 - solution
Identifying a pattern in a list of numbers
using addition
Since the numbers are increasing
relatively slowly, try addition.
What number comes next?
3, 8, 13, 18, ___
3,
8,
+5
5
13,
+5
18,
+5
___
+5
The common difference between each pair of
numbers is 5.
Therefore, the next number is 18 + 5 = 23.
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1
Example 2
Example 2 - Solution
Identifying a pattern in a list of numbers using
multiplication
Since the numbers are increasing relatively
quickly, try multiplication.
3,
x3
What number comes next?
3, 9, 27, 81, ___
9,
27,
x3
81,
x3
___
x3
The common ratio between each pair of
numbers is 3. Thus, the next number is:
81 x 3 = 243.
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Example 3
Fibonacci Sequence
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Example 4 - Predicting the next figure in a
sequence by finding the pattern
What comes next in this list of numbers?
1, 1, 2, 3, 5, 8, 13, 21, ?
,
Solution: This pattern is formed by adding
the previous 2 numbers to get the next
number:
So the next number in the sequence is:
13 + 21 = 34
,
, ...
What is the next figure in the sequence?
Solution:
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Example 5 – find the next figure in
the pattern
,
,
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Inductive Reasoning:
More than one Solution!
Is this illustration a
wine goblet or
two faces looking
at each other?
,...
Solution:
11
12
2
Inductive Reasoning:
More than one Solution!
Deductive Reasoning
2, 4, ?
What is the next number in this sequence?
The process of drawing a specific
conclusion from one or more general
statements.
A theorem is a conclusion proved true by
deductive reasoning
If the pattern is to add 2 to the previous
number it is 6.
If the pattern is to multiply the previous
number by 2 then the answer is 8.
We need to know one more number to
decide.
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Example 5
An Example in Everyday Life
Everyday
Situation
The ad says “All
Jewelry On Sale
- 50% off”.
You want to
buy a ring. You
only pay $50
for a ring
normally priced
at $100.
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Example 6
Deductive Reasoning
Using Inductive Reasoning, apply the rules to
specific numbers. Do you see a pattern?
General Statement:
All jewelry is 50% off.
A ring is jewelry.
Conclusion:
A ring originally $100
is sale priced at $50.
Select any
number
4
7
11
4 +3 = 7
7 + 3 = 10
11 +3 = 14
Multiply by 4
7 x 4 = 28
10 x 4 = 40
14 x 4 = 56
Divide this
product by 2
28
= 14
2
14 – 6 = 8
40
= 20
2
56
= 28
2
28 – 6 = 22
Add 3
Subtract 6 from
the quotient
15
20 – 6 = 14
The output number is twice the input number
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Example 6 continued
Using Deductive reasoning:
Select any number
Add 3
Multiply by 4
Divide by 2
Subtract 6
n
n+3
(n + 3) x 4
(n + 3) × 4 4n + 12
=
= 2n + 6
2
2
2n + 6 – 6 = 2n
Which agrees with our inductive hypothesis.
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