# Difference between revisions of "True or false"

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Oscarlevin (talk | contribs) (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. | ||

+ | |||

+ | |||

+ | {{needs_hint}} | ||

+ | {{needs_answer}} | ||

+ | {{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. | ||

+ | |||

+ | }} | ||

[[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.

- Statement 5 is false.
- Statement 1 is false.
- Statement 4 is true.
- Only one of these statements is false.
- 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.