True or false

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.