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
6
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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Assignment