Transcript Section 1.1 Slides

Chapter 1
Section 1-1
Types of Reasoning
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Characteristics of Inductive and
Deductive Reasoning
Inductive Reasoning
• Use specific premise(info) to draw a general
conclusion (a conjecture) from repeated
observations.
• May or may not be true.
Deductive Reasoning
• Apply general principles to draw specific
conclusions.
• Absolute truth
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Example: determine the type of
reasoning
Determine whether the reasoning is an example
of deductive or inductive reasoning.
All math teachers have a great sense of humor.
Patrick is a math teacher. Therefore, Patrick
must have a great sense of humor.
Solution
Because the reasoning goes from general to
specific, deductive reasoning was used. No
other conclusion can be deduced.
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Example: determine the type of
reasoning
Determine whether the reasoning is an example
of deductive or inductive reasoning.
Carrie’s first three children were boys. If she
has another child, it will be a boy.
Solution
Because the reasoning goes from specific to
general, inductive reasoning was used. She
may also have a girl.
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Example: find next term
Use inductive reasoning to determine the
probable next number in the list below.
1, 1, 2, 3, 5, 8, ____ (Fibonacci sequence)
Use inductive reasoning to determine the
probable next number in the list below.
O, T, T, F, F, S, S, E, N, _____
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Example: find next term
Use inductive reasoning to determine the
probable next number in the list below.
1, 3, 5, 7, ____ (arithmetic sequence)
Use inductive reasoning to determine the
probable next number in the list below.
2, 4, 8, 16, _____ (geometric sequence)
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Example: predict the product of two
numbers
Use the list of equations and inductive reasoning
to predict the next multiplication fact in the list:
37 × 3 = 111
37 × 6 = 222
37 × 9 = 333
37 × 12 = 444
Solution
37 × 15 = 555
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Example: predicting the next number
in a sequence
Use inductive reasoning to determine the
probable next number in the list below.
2, 9, 16, 23, 30
Solution
Each number in the list is obtained by adding 7
to the previous number.
The probable next number is 30 + 7 = 37.
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Pitfalls of Inductive Reasoning
One can not be sure about a conjecture until a
general relationship has been proven.
One counterexample is sufficient to
make the conjecture false.
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Example: pitfalls of inductive
reasoning
We concluded that the probable next number
in the list 2, 9, 16, 23, 30 is 37.
If the list 2, 9, 16, 23, 30 actually represents
the dates of Mondays in June, then the date of
the Monday after June 30 is July 7 (see the
figure on the next slide). The next number on
the list would then be 7, not 37.
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Example: pitfalls of inductive
reasoning
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Example: use deductive reasoning
Find the length of the hypotenuse in a right
triangle with legs 3 and 4. Use the Pythagorean
Theorem: c 2 = a 2 + b 2, where c is the
hypotenuse and a and b are legs.
Solution
c 2 = 32 + 4 2
c 2 = 9 + 16 = 25
c = 5 (absolute truth – general to specific)
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Example: Use the method of Gauss
Find the sum
1+2+3+……+798+799+800.
Solution
400 sums of 801 = 400(801)=320,400
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Example: Use the method of Gauss
Modify the method of Gauss to find the sum
2+4+6+……+100.
Solution
Factor out 2 first
2 times 25 sums of 51 =2(25)(51)
Test question
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Example: Use the method of Gauss
Modify the method of Gauss to find the sum
8+13+18+23+……43+48+53.
Solution
Note: the difference is 5
8+53=61
13+48=61
5(61)=305
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