Transcript 2.3 Deductive Reasoning

Inductive and Deductive
Reasoning
Objectives
• Familiarize you with the deductive reasoning
process
• Learn the relationship between inductive and
deductive reasoning
Problem Solving
 Logic – The science of correct reasoning.
 Reasoning – The drawing of inferences or
conclusions from known or assumed facts.
 When solving a problem,
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one must understand the question,
gather all pertinent facts, analyze the problem
i.e. compare with previous problems
(note similarities and differences),
perhaps use pictures or formulas to solve the
problem.
Deductive vs. Inductive Reasoning
 The difference:
inductive reasoning uses patterns to arrive at a
conclusion (conjecture)
deductive reasoning uses facts, rules, definitions or
properties to arrive at a conclusion.
Deductive reasoning is the
process of reasoning logically
from given statements to a
conclusion.
Examples of Inductive Reasoning
1) Every quiz has been easy.
Therefore, the test will be easy.
2) The teacher used PowerPoint in the
last few classes. Therefore, the
teacher will use PowerPoint
tomorrow.
3) Every fall there have been hurricanes
in the tropics. Therefore, there will be
hurricanes in the tropics this coming
fall.
Example of Deductive Reasoning
An Example:
The catalog states that all entering
freshmen must take a mathematics
placement test.
You are an entering freshman.
Conclusion: You will have to take a
mathematics placement test.
Inductive or Deductive Reasoning?
Geometry example…
x
Triangle sum property –
the sum of the angles of any
triangle is always 180 degrees.
Therefore, angle x = 30°
60◦
Deductive Reasoning – conclusion is based on a
property
Inductive or Deductive Reasoning?
Geometry example…
What comes next?
Is there a rule?
Colored triangle rotating 90°
CW in the corners of the square
Inductive Reasoning
Who is known for
using Deductive
Reasoning?
Sherlock Holmes
Sherlock Holmes would use deductive reasoning to
help solve crimes.
Example:
Sherlock Holmes could help solve a
mystery by making inferences.
If Holmes saw a pack of cigarettes by
a victim (the victim did not smoke),
Holmes can make the assumption
that the killer is a smoker.
Sherlock Holmes
Andrew Ault
However…
 Deductive reasoning may not be the most accurate way
of solving a problem, because we all know that
assumptions can be wrong.
Monty Python
How Can Deductive Reasoning Be
Applied In School?
Math
Scott has a case of soda in his house since there are 13
cans of soda left I deduce that Scott has drank 11 cans of
soda.
English
When I see ‘like’ in a sentence, I deduce that it is a simile.
Science
Using laws and rules to make assumptions
The law of gravity means everything that goes up must
come down
I threw a baseball in the air
That means the baseball must come down.
Social Studies
To be elected President you must obtain at least 270
electoral votes
George Bush won 287 electoral votes
Therefore George Bush is the President.
What professions do you think
commonly use Deductive
Reasoning?
Why?
 These professions tend to ask a lot of questions to
try to solve problems or to prove a point.
 Often they would have to make assumptions to
solve problems.
 They would use rules and widely accepted beliefs to
prove their argument.
For example:
An attorney states that his client is innocent because the
crime victim was hit by a car. Since his client does not have a
license. He can deduce that his client is innocent.
An auto mechanic knows that if a car
has a dead battery, the car will not
start. A mechanic begins work on a
car and finds the battery is dead.
What conclusion will she make?
The mechanic can conclude that
the car will not start.
Using Inductive Reasoning to Find
a Pattern (yay Math!)
 1, 2, 4, 5, 7,8,10, 11, 13, ___, ___
 Solution: the pattern seems to be to add 1, then add 2,
then add 1, then add 2,…
 0,1,1,2,3, 5, 8, 13, 21 ___, ___
 Solution: It’s the Fibonacci Sequence, one of the most
famous mathematical patterns! The pattern is after the
0 and the 1, you add the previous 2 numbers to get the
next number. 0+1=1, 1+1=2 1+2=3…
Using Inductive Reasoning to Make a
Conjecture (educated guess)
 Select a number:
 Add 50:
 Multiply by 2:
 Subtract the original number:
 Result:
 Now Prove it Using Deductive Reasoning!
Using Inductive Reasoning to Make a
Conjecture (educated guess)
 Select a number:
 Add 50:
 Multiply by 2:
 Subtract the original number:
 Result:
x
x + 50
2(x+50) or 2x + 100
2x + 100 - x
X + 100
Deductive Reasoning
 This method of reasoning produces results that are
certain within the logical system being developed.
 It involves reaching a conclusion by using a formal
structure based on a set of undefined terms and a set
of accepted unproved axioms or premises.
 The conclusions are said to be proved and are called
theorems.
Deductive Reasoning
 For any given set of premises, if the conclusion is
guaranteed, the arguments is said to be valid.
 If the conclusion is not guaranteed (at least one
instance in which the conclusion does not follow),
the argument is said to be invalid.
 BE CARFEUL, DO NOT CONFUSE TRUTH WITH
VALIDITY!
Deductive Reasoning
Examples:
1.
All students eat pizza.
Claire is a student at CSULB.
Therefore, Claire eats pizza.
2. All athletes work out in the gym.
Barry Bonds is an athlete.
Therefore, Barry Bonds works out in the gym.
Deductive Reasoning
3. All math teachers are over 7 feet tall.
Miss Sweeney is a math teacher.
Therefore, Miss Sweeney is over 7 feet tall.
 The argument is valid, but is certainly not true.
Examples
 No one who can afford health insurance is
unemployed.
All politicians can afford health insurance.
Therefore, no politician is unemployed.
VALID OR INVALID?????
The argument is valid
Finding a Counter Example to
prove an argument wrong…
 Prove the statement is false:
 A number is divisible by 3 if it’s last two digits are
divisble by 3.
 Counter Examples: 1527, 136, 418, and many more…
 The sum of any three odd numbers is even.
 Counter Examples: 1+1+1=3, 1+3+1=5, 1+3+7+11,and
many more…