2.3 Deductive Reasoning

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2.3 Deductive Reasoning

• p. 87

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Reminders

• Statement
• Conditional statement
• Converse
• Inverse
• Contrapositive
• Biconditional

• Symbols
• p ? q
• q ? p
• p ? q
• q ? p
• p ? q

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Ex Given p it is 4th periodq it is time
for lunch

• Write p?q.
• If it is 4th period, then it is time for lunch.
• Write p.
• It is not 4th period.
• Write p?q.
• It is 4th period iff it is time for lunch.
• Is p?q true?

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Laws of Deductive Reasoning

• 1. Law of Detachment
• 2. Law of Syllogism

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Law of Detachment

• If a statement p?q is given and a second
  statement p is given, then a third statement q
  results.
• Given p?q
• p
• q
• Ex 1. If x is even, then x2 is even.
• 2. x 6 What statement follows?
• 62 is even

q
p
p
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More examples
p
q

• Given 1. If it is raining, then the ground is
  wet.
• 2. It is raining. What
  follows?
• The ground is wet.
• Given 1. If an lt is between 0o and 90o, then it
  is acute.
• 2. ltB is acute. What follows?
• No conclusion.

p
p
q
q
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Law of Syllogism

• If p?q is given and q?r is given, then p?r
  results.
• Given p?q
• q?r
• p?r

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Example
p

• Given 1. If Tony is sick on Friday, then he
  cannot play football.
• 2. If Tony cannot play football, then the
  team will lose.
• What statement follows?
• If Tony is sick on Friday, then the team will
  lose.

q
q
r
p?q q?r p?r
p
r
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Example

• Given p?q
• q?s
• r?s
• r?q
• What follows?
• No conclusion.

• Given q?r
• s?t
• r?s
• p?q
• What follows?
• p?t

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EXAMPLE 1
Use the Law of Detachment
Use the Law of Detachment to make a valid
conclusion in the true situation.
SOLUTION
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EXAMPLE 1
Use the Law of Detachment
Today is Friday satisfies the hypothesis of the
conditional statement, so you can conclude that
Mary will go to the movies tonight.
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EXAMPLE 2
Use the Law of Syllogism
If possible, use the Law of Syllogism to write a
new conditional statement that follows from the
pair of true statements.
If x gt 5, then x2 gt 25.
If a polygon is regular, then all of its sides
are congruent.
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EXAMPLE 2
Use the Law of Syllogism
SOLUTION
If Rick takes chemistry this year, then Rick will
get an A in chemistry.
If x gt 5, then x2 gt 20.
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EXAMPLE 2
Use the Law of Syllogism
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for Examples 1 and 2
GUIDED PRACTICE
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for Examples 1 and 2
GUIDED PRACTICE
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for Examples 1 and 2
GUIDED PRACTICE
State the law of logic that is illustrated.
If you get an A or better on your math test, then
you can watch your favorite actor.
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for Examples 1 and 2
GUIDED PRACTICE
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EXAMPLE 3
Use inductive and deductive reasoning
ALGEBRA What conclusion can you make about the
product of an even integer and any other integer?
SOLUTION
STEP 1
Look for a pattern in several examples. Use
inductive reasoning to make a conjecture.
(2) (2)
4,
(1) (2)
2,
4,
3 (2)
6,
2 (2)
(1) (4)
4,
12
(2) (4)
8,
2 (4)
8,
3 (4)
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EXAMPLE 3
Use inductive and deductive reasoning
STEP 2
Let n and m each be any integer. Use deductive
reasoning to show the conjecture is true.
2n is an even integer because any integer
multiplied by 2 is even.
2nm represents the product of an even integer and
any integer m.
2nm is the product of 2 and an integer nm. So,
2nm is an even integer.
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EXAMPLE 4
Reasoning from a graph
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EXAMPLE 4
Reasoning from a graph
SOLUTION
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for Examples 3 and 4
GUIDED PRACTICE
SOLUTION
Conjecture The sum of a number and itself is
twice the number.
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for Examples 3 and 4
GUIDED PRACTICE
SOLUTION
Using inductive reasoning The more strokes it
takes for the northern elephant to surface, the
deeper it dove.
Using deductive reasoning The northern elephant
seal uses fewer strokes to surface from 190
meters then from 410 meters.
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Assignment
pp. 91-94 8-25, 30-35, 45-51