Section 2.3: Deductive Reasoning

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Section 2.3 Deductive Reasoning
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Goals

• Use symbolic notation to represent logical
  statements
• Form conclusions by applying the laws of logic to
  true statements

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Symbolic Notation

• Conditional Statement
• p ? q
• Converse
• q ? p
• Inverse
• p ? q
• Contrapositive
• q ? p
• Biconditional
• p ? q

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Example 1 Using Symbolic Notation

• Let p be the value of x is -4 and q be the
  square of x is 16.
• Write p ? q in words
• If the value of x is -4, then the square of x is
  16
• Write q ? p in words
• If the square of x is 16, then the value of x is
  -4
• Decide whether the biconditional statement p ? q
  is true
• the conditional statement in part (a) is true,
  but the converse in part (b) is false

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Example 2 Writing an Inverse and a Contrapositive

• Let p be today is Monday and q be there is
  school.
• a) Write the contrapositive of p ? q.
• q ? p, if there is no school, then today is
  not Monday
• b) Write the inverse of p ? q
• p ? q, If today is not Monday, then there is
  no school

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Using The Laws of Logic

• Inductive Reasoning
• Looking at several specific situations to arrive
  at a conjecture
• Deductive Reasoning
• Uses a rule to make a conjecture

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• Logical Argument
• Using facts, definitions, and accepted properties
  in a logical order
• Laws of Deductive Reasoning
• Law of Detachment
• Law of Syllogism

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Law of Detachment

• If p ? q is a true statement and p is true, then
  q is true

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Law of Syllogism

• If p ? q and q ? r are true conditionals, then p
  ? r is also true