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True or false

Revision as of 18:57, 13 October 2010 by Zerrakhi (talk | contribs)
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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.