Download Inductive and Deductive Reasoning

Inductive and Deductive Reasoning
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Definition: Conjecture:
an unproven statement that is based on observations or given
information.
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Definition: Counterexample:
a specific case for which a conjecture is false.
Counterexample
Find a counter example to show that the following
conjecture is false.
The sum of two numbers is always greater than the larger
number.
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Definitions: Conditionals, Hypothesis, &
Conclusions:
A conditional statement is a logical statement that has two
parts: If ____ then _____.
The hypothesis is the “if ” part and it tells you what you are
talking about.
The conclusion is the “then” part and it describes the
hypothesis.
Writing a conditional statement
 Writing the following statements as conditionals.
Two angles that make a linear pair are supplementary.
All 90o angles are right angles.
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 The negation of a statement is the opposite of the original.
Negation
 Negate the following statements.
The ball is red.
The cat is not black.
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Definitions: Inverse, Converse, Contrapositive
 The converse of a conditional statement switches the
hypothesis and conclusion.
 The inverse of a conditional statement negates both the
hypothesis and conclusion
 The contrapositive of a conditional statement takes the
inverse of the converse. (it switches and negates)
Writing statements
 Write the converse, inverse and contrapositive of the
conditional statement:
“If two angles form a linear pair, then they are
supplementary.”
Which of these statements are true?
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Definition: Biconditional:
If a conditional statement and its converse are both true, then
we can write it as a biconditional statement by using the phrase
if and only if instead of putting it in if-then form.
__________ if and only if ___________.
(hypothesis)
(conclusion)
Biconditional Statement
 Write the following conditional statement as a
biconditional statement.
 If two lines intersect to form a right angle, then they are
perpendicular.
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The Law of Detachment
This applies when one statement is conditional and a second
statement confirms the hypothesis of the conditional. The
conclusion is then confirmed.
Here is an example.
Deductive Reasoning
If it is Friday, then Mary goes to the movies.
It is Friday.
What conjecture can you make from the above statements?
Deductive Reasoning
 If two angles form a linear pair, then they are supplementary.
 Angle 1 and Angle 2 are a linear pair.
Deductive Reasoning
 If two angles form a linear pair, then they are supplementary.
 Angle 1 and Angle 2 are supplementary.
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The Law of Syllogism
 This applies when you have two conditional statements. The
conclusion of one, confirms the hypothesis of the other. In
this case our result is still a conditional with the first
hypothesis and the second conclusion.
(I call this the “Oreo Cookie” Law.) Here is how it works…
Deductive Reasoning
If it is Friday, then Mary goes to the movies.
If Mary goes to the movies then she gets popcorn.
 Combine the two above conditional statements into one
conditional statement.
Deductive Reasoning
 If two angles form a linear pair, then they are supplementary.
 If two angles are supplementary then their sum is 180
degrees.
Deductive Reasoning
 If a polygon is regular, then all angles in the interior of the
polygon are congruent. If a polygon is regular, then all of its
sides are congruent.
 Why can’t these two statements be combined like the last
example.