Inductive and Deductive Reasoning (1).ppt

Inductive and Deductive Reasoning

Inductive Reasoning
The type of reasoning that forms a conclusion based on
the examination of specific examples. The conclusion formed
by using inductive reasoning is often called a conjecture,
since it may or may not be correct.
Inductive reasoning is the process of reaching a general
conclusion by examining specific examples.

Example of Inductive Reasoning to
predict a number
3, 6, 9, 12, 15
The pattern is
a difference of
3
1, 3, 6, 10, 15
The pattern starts
with a difference of
2 then
continuously add
one

Example of Inductive Reasoning to
predict a number
5, 10, 15, 20, 25 2, 5, 10, 17, 26,
The pattern is a difference
of 5 so the next number is
30.
The pattern starts with
a difference of 3 then
continuously
add 2. Next number is 37

Use Inductive Reasoning
to Make a Conjecture
Consider the following procedure: Pick a number. Multiply
the number by 8, add 6 to the product, divide the sum by 2, and
subtract 3. Complete the above procedure for several
different numbers. Use inductive reasoning to make a conjecture
about the relationship between the size of the resulting number
and the size of the original number.
03

Solution
Original number: 5
Multiply by 8: 8 x 5 = 40
Add6: 40 + 6 = 46
Divide by 2: 46 / 2 = 23
Subtract 3: 23 - 3 = 20
We started with 5 and followed the procedure to produce 20.
Starting with 6 as our original number produces a final result of 24.
Starting with 10 produces a final result of 40. Starting with 100 produces
a final result of 400. In each of these cases the resulting number is four
times the original number

PENDULUM
Scientists
instance,
inductive
often use inductive reasoning. For
used
Galileo Galilei (1564– 1642)
reasoning to discover that the time
required for a pendulum to complete one swing,
called the period of the pendulum, depends on the
length of the pendulum. Galileo did not have
a clock,
pendulums
so he measured the periods of
in “heartbeats.” The following table
shows some results obtained for pendulums of
various lengths. For the sake of convenience, a
length of 10 inches has been designated as 1 unit.

If a pendulum has a length of
49 units, what is its period?
If the length of a pendulum is
quadrupled, what happens to
its period?
a pendulum with a length of 4 units has a period that is
twice that of a pendulum with a length of 1 unit. A
pendulum with a length of 16 units has a period that
EXAMPLE
is twice that of a
It appears that
pendulum with a length of 4 units.
quadrupling the length of a
pendulum doubles its period.

EXAMPLE
A tsunami is a sea wave produced by
an underwater earthquake. The height
of a tsunami as it approaches
land depends on the velocity of the
tsunami. Use the table at the left and
inductive reasoning to answer each of
the following questions.

EXAMPLE
What happens to the height of a tsunami
when its velocity is doubled? the height
of a tsunami quadrupled if
its speed is doubled.
What should be the height of a tsunami if
its velocity is 30 feet per second? 100 ft

EXAMPLE
Conclusions based on
inductive reasoning may
be incorrect. As an
illustration, consider the
circles shown below.
For each circle, all
possible line segments
have been drawn to
connect each dot on the
circle with all the other
dots on the circle.

EXAMPLE
There appears to be a pattern. Each additional
dot seems to double the number of regions. Guess the
maximum number of regions you expect for a
circle with six dots. Check your guess by counting the
maximum number of regions formed by the line
segments that connect six dots on a large circle.
Your drawing will show that for six dots, the maximum
number of regions is 31 (see the fi gure at
the left), not 32 as you may have guessed. With seven
dots the maximum number of regions is 57. This is a
good example to keep in mind.
Just because a pattern holds true for a few
cases, it does not mean the pattern will continue. When
you use inductive reasoning, you have no guarantee
that your conclusion is correct.

Counterexamples
A statement is a true statement provided that it is
true in all cases. If you can find one case
for which a statement is not true, called a
counterexample, then the statement is a false

Example of Counterexample
[x] > 0
[0] > 0
0 > 0 false
√x² = x
√(-3 ) ² = -3 false
x²> x
1² > 1
1 > 1 false

x
x
= 1
x + 3 = x + 1
3
√x² + 16 = x + 4

0
0
= 1
3 + 3 = 3 + 1
3
Indeterminate 2 = 4
√3 ² + 16 = 3 + 4
Sqrt of 25 = 7

Deductive Reasoning
Deductive reasoning is distinguished from inductive reasoning in
that it is the process of reaching a conclusion by
applying general principles and procedures.

Example
Procedure: Pick a number. Multiply the number by 8, add 6
to the product, divide the sum by 2, and subtract 3.
Let n represent the original number. n
Multiply the number by 8:
Add 6 to the product:
8n
8n + 6
Divide the sum by 2: 8n + 6
= 4n + 3
2
4n + 3 - 3 = 4n
Subtract 3:

Example
Procedure: Pick a number. Multiply the number by 6, add 10
to the product, divide the sum by 2, and subtract 5.
Let n represent the original number.: n
Multiply the number by 6:
Add 10 to the product:
6n
6n + 10
Divide the sum by 2: 6n + 10
= 3n + 5
2
3n + 5 - 5 = 3n
Subtract 5:

Inductive Reasoning
vs.
Deductive Reasoning

Inductive Reasoning
During the past 10 years, a tree has produced plums
every other year. Last year the tree did not produce
plums, so this year the tree will produce plums.
This argument reaches a conclusion based on
specific examples, so it is an example of inductive
reasoning

Inductive Reasoning
I know I will win a jackpot on this slot machine in the next 10
tries, because it has not paid out any money during the last
45 tries.
This argument reaches aconclusion based on specific
examples, so it is an example of inductive reasoning.

Deductive Reasoning
All home improvements cost more than the estimate. The
contractor estimated that my home improvement will cost
$35,000. Thus my home improvement will cost more than
$35,000.
Because the conclusion is a specific case of a general
assumption, this argument is an example of deductive
reasoning.

Deductive Reasoning
All Janet Evanovich novels are worth reading.
The novel Twelve Sharp is a Janet Evanovich
novel. Thus Twelve Sharp is worth reading.
Because the conclusion is a specific case of
a general assumption, this argument is an
example of deductive reasoning.

“There is always a reason for
every decision that we make.
Inductive or deductive… doesn’t
matter as long as it is the right
decision”
-Godofredo Trajano Tesorio

Inductive and Deductive Reasoning (1).ppt

More Related Content

Similar to Inductive and Deductive Reasoning (1).ppt (20)

Recently uploaded (20)

Inductive and Deductive Reasoning (1).ppt