Section 2.3
Introduction to Deductive
Reasoning
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Inductive Reasoning
Inductive reasoning is the process of
forming conclusions on the basis of
patterns and observations.
Example:
Observation: I notice that a student in my
class has never handed in homework.
Conclusion: The student will not hand in
homework in the future.
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Deductive Reasoning
In deductive reasoning, we form a
conclusion based on one or more given
statements. The conclusion must be true
if the statements (premises) are true.
Example:
Statements:
1) The product of two whole numbers is 35.
2) One of the numbers is 7.
Conclusion:
The other number is 5
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Conditional Statement
The given information in deductive
reasoning is often given in the form of a
conditional statement ( “If…., then,…”).
Example: If a number is greater than 10,
then it is greater than 8.
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4) Write the following in if-then form:
a) The parade will be on Thursday if
Flag Day is on Thursday.
b) People under 16 years of age
cannot obtain a driver’s license.
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6) Draw a Venn diagram to
illustrate each statement.
a) Every Member of Hillsville 500 Club
is an alumnus of Hillsville High
School.
b) A tree over the Forest Service’s size
limit will not be cut.
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Converse, Inverse, Contrapositive
Statement:
If p, then q.
Converse:
If q, then p.
Inverse:
If not p, then not q.
Contrapositive: If not q, then not p.
Example: Write the converse, inverse, and
contrapositive of the following statement.
“If an animal is a horse, then it has four legs.”
Is the converse true? Inverse? Contrapositive?
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Contrapositive
A conditional statement and its
contrapositive are logically equivalent.
If one is true, so is the other. If one is
false, so is the other.
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Law of Detachment
Premises
1. If p, then q.
2. p
Conclusion: q (valid)
Law of Contraposition
Premises
1. If p, then q.
2. Not q.
Conclusion: Not p (valid)
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Use the law of detachment or the law of
contraposition to form a valid conclusion
from each set of premises. Draw a Venn
diagram to support your conclusion.
24) Premises: If poison is present in the
bone marrow, then production of red
blood cells will be slowed down. The
patient has poison in her bone marrow.
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If a conditional statement and its converse
are true, we can write a new statement
called a biconditional statement using the
words if and only if.
16)Combine the statement and its converse
into a biconditional statement.
“If Smith is guilty, then Jones is innocent.
If Jones is innocent, then Smith is guilty.”
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18) Write the biconditional statement as
two separate statements -- a
conditional statement and its
converse.
“There will be negotiations if and
only if the damaged equipment is
repaired.”
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