Transcript Inductive and Deductive Reasoning

Logical Arguments in Mathematics
A proof is a collection of statements and
reasons in a logical order used to verify
universal truths.
However… depending upon the type of
proof the definition can and will change.
Deductive Proof
 Step by step process of drawing conclusions based on
previously known truths.
 Properties of Deductive Proofs:
 Uses “Top Down” Logic and Reasoning
 Takes a general statement made about an entire class of
things and then applies the rule to one specific example.
 Only acceptable form of a proof (Scientific and
mathematical)
Deductive Reasoning: Flow Chart
Hypothesis
• Postulates
• Theorems
• Definitions
Theory
• Assumptions
• Testing
Confirmation
• Specific Case
• Proof
Observation
• Conclusion
• Specific Case
Definitions used in Deductive Arguments
 Logical Statements: Statements that can be true or false.
 In logical analysis, variables no longer represent numbers…
instead they represent logical statements.
 Most logical statements are written as conditional
statements.
 Example:
 p – Paris is the capital of France
 q – The moon is made of green cheese
 What does the conditional statement: If p, then q say?
Conditional Statements
 Deductive Arguments are based on conditional
statements.
 All the postulates and theorems we are studying are
conditional statements.
 When proving a theorem… we assume the hypothesis
and show how to get the conclusion.
 For a conditional statement to be true consider the
following:
Using Conditional Statements to
Complete Deductive Proofs
 Look for the assumption of the hypothesis
 Follow each piece of the argument carefully.
 Remember… very similar to the transitive property!
 Example:
 If Lyn is taller than Mark, then Mark is taller than Eddie.
 Lyn is taller than Mark.
 What can you conclude about Mark?
Problems with Deductive Arguments
 Errors in deductive arguments are called fallacies.
 Examine the following argument. Why might it not be
a “good” argument?
 Premise:
 All good basketball players are over 6 feet tall.
 Grant is 6 foot 3 inches tall.
 Conclusion:
 Grant is a good basketball player.
Practice with Deductive Arguments
1.
When the sun shines, the grass grows. When the grass grows, it
needs to be cut. The sun is shining. What can you deduce about
the grass?
2.
Jim is a good barber. Everybody who gets a haircut by Jim gets a
good haircut. Austin has a good haircut. What can you deduce
about Austin?
3.
Why is the following example of deductive reasoning faulty?
Premise:
Khaki pants are comfortable
Comfortable pants are expensive
Adrian’s pants are not khaki pants
Conclusion: Adrian’s pants are not expensive
Logical Arguments in Mathematics and Real Life
 Examine the following argument. Explain how this
argument is different. Is the conclusion of the
argument true?
 Argument 1:
 After picking roses for the first time, Jamie began to
sneeze. She also began sneezing the next four times she
was near roses. Based on these past experiences, Jamie
decides that she is allergic to roses.
Inductive Proof
 The process of arriving at a conclusion based in a set of
observations.
 Properties of Inductive Proofs:
 Uses “Bottom Up” Logic and Reasoning
 Highly based on patterns
 Takes specific incidents of an event to develop an overall
conclusion
 Downfall… NOT an acceptable form of proof 
Inductive Reasoning: Flow Chart
Major Problems with Inductive
Arguments
 Since many inductive arguments are based on
patterns, there is NO guarantee that the conditions
will always be true.
 Example:
 The number pi… originally it was thought that pi had an
exact value , i.e recognizable pattern.
Benefits to Inductive Arguments
 A hypothesis based on inductive reasoning can lead to
a more careful study of a situation.
 Allows for more in-depth development of hypotheses
for experiments.
 Many times theories in science, mathematics, and
education are developed and tested using inductive
arguments
Examples of Induction
 Numerical Patterns: Find the next two terms of each
sequence
 1, 4, 16, 64, …
,
How?
 18, 15, 12, 9, …
,
How?
 10, 12, 16, 22, …
,
How?
 8, -4, 2, -1, ½,…
,
How?
 2, 20, 10, 100, 50…
,
How?
 Extra Credit:
 Write the equations to represent each of the sequences above.
Inductive or Deductive?
 Examine the following scenarios. Determine if the
arguments use deductive or inductive reasoning.
 Argument 1:
 Jake noticed that spaghetti has been on the school menu
for the past five Wednesdays. Jake decides that the
school always serves spaghetti on Wednesday.
 Argument 2:
 By using the definitions of equilateral triangles and of
perimeter, Katie concludes that the perimeter of every
equilateral triangle is three times the length of a side.
Inductive or Deductive?
 Argument 3:
 Brendan observes that (-1)2 = +1; (-1)4 = +1; and (-1)6 = +1. He
concludes that every even power of (-1) equals +1
 Argument 4:
 There are three sisters. Two of them are athletes and two of
them like ice cream. Can you be sure that both of the
athletes like ice cream.
 Do you reason deductively or inductively to conclude the
following: At least one of the athletic sisters like ice cream?