Difference between revisions of "True or false"

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(Created page with 'Assign a truth value to each of the following statements. # Statement 5 is false. # Statement 1 is false. # Statement 4 is true. # Only one of these statements is false. # Statem…')
 
 
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# Only one of these statements is false.
 
# Only one of these statements is false.
 
# Statements 2 and 3 are both true or both false.
 
# Statements 2 and 3 are both true or both false.
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{{needs_answer}}
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{{solution |
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First identify statements with equal vs opposite truth values:
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* 1 has opposite truth value to 5 (from statement 1)
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* 2 has opposite truth value to 1 (from statement 2), hence equal truth value to 5
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* 3 has equal truth value to 4 (from statement 3)
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* 5 true only if 2 has equal truth value to 3 (from statement 5)
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If 4 false, then:
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*3 false (equal value to 4)
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*2 has opposite value to 5 (5 true only if 2 false, given 3 false)
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But 2 has equal truth value to 5.
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So 4 true.
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If 4 true, then:
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*3 true (equal value to 4)
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*2 and 5 true (otherwise they'd both be false, contradicting 4)
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*1 false (opposite value from 2, 5)
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Hence 1 is false and all other statements are true.
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}}
  
 
[[Category: Liar puzzles]]
 
[[Category: Liar puzzles]]
 
[[Category: Logic]]
 
[[Category: Logic]]

Current revision as of 18:57, 13 October 2010

Assign a truth value to each of the following statements.

  1. Statement 5 is false.
  2. Statement 1 is false.
  3. Statement 4 is true.
  4. Only one of these statements is false.
  5. Statements 2 and 3 are both true or both false.
Solution

First identify statements with equal vs opposite truth values:

  • 1 has opposite truth value to 5 (from statement 1)
  • 2 has opposite truth value to 1 (from statement 2), hence equal truth value to 5
  • 3 has equal truth value to 4 (from statement 3)
  • 5 true only if 2 has equal truth value to 3 (from statement 5)

If 4 false, then:

  • 3 false (equal value to 4)
  • 2 has opposite value to 5 (5 true only if 2 false, given 3 false)

But 2 has equal truth value to 5. So 4 true.

If 4 true, then:

  • 3 true (equal value to 4)
  • 2 and 5 true (otherwise they'd both be false, contradicting 4)
  • 1 false (opposite value from 2, 5)

Hence 1 is false and all other statements are true.