Section 3'4 Truth Tables for the Conditional and Biconditional

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Section 3.4 Truth Tables for the Conditional and
Biconditional

• Thinking Mathematically
• Math 1001

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Objectives

• Understand the logic behind the definition of the
  conditional.
• Construct truth tables for conditional
  statements.
• Write a conditional statement in an equivalent
  form.
• Use a truth table to determine if a compound
  statement is a tautology.

• Understand the definition of the biconditional.
• Construct truth tables for biconditional
  statements.
• Construct truth tables for statements consisting
  of three simple statements.
• Determine the truth value of a compound statement
  for a specific case.

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Conditional Statements, ?

• If you pass the final, then you pass the course.
• p You pass the final.
• q You pass the course.

T
F
T
T
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The Definition of the Conditional (page 116)

• In the conditional
• p ? q
• p is the antecedent
• q is the consequent
• antecedent ? consequent
• A conditional is false only when the antecedent
  is true and the consequent is false.

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Truth table for q ? p
T
F
T
T
Only false when antecedent is true and the
consequent is false.
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A Statement that is Equivalent to the Conditional

• If you pass the final, then you pass the course.
• p You pass the final.
• q You pass the course.
• p ? q

• If you do not pass the course, then you did not
  pass final.
• p You pass the final.
• q You pass the course.
• q ? p

The Truth value of a conditional statement does
not change if the antecedent and the consequent
are reversed and negated.
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Truth table for (pVq)p ? q
T
A tautology is a compound statement that is
always true.
T
T
T
Only false when antecedent is true and the
consequent is false.
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Definition of Tautology (page 118)

• A tautology is a compound statement that is
  always true.
• To determine if a compound statement is a
  tautology construct a truth table.
• If the last column is always true then the
  statement is a tautology.
• Conditional statements that are tautologies are
  called implications.

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Biconditional Statements

• ? translates into if and only if
• p ? q means p?q and q?p
• p?q q?p

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Truth table for (p?q) (q?p)
T
T
T
F
F
F
T
T
Only false when antecedent is true and the
consequent is false.
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Example 6 Constructing a Truth Table with Eight
Cases (page 121)
T
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
Only false when antecedent is true and the
consequent is false.
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Section 3.4 Homework

• 1 60

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Construct a truth table for p ? q
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2. Construct a truth table for p ? q
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5. Construct a truth table for (p q) ? (p V q)
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21. Construct a truth table for (p q) (p V
q)
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Four Case Problem
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42. (p V q) ? r
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Eight Case Problem