Exclusive Or/Examples/One of Five Statements is True
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Example of Exclusive Or
Which of the following $5$ statements is true?
- $(1): \quad$ Exactly one of these statements is false.
- $(2): \quad$ Exactly two of these statements are false.
- $(3): \quad$ Exactly three of these statements are false.
- $(4): \quad$ Exactly four of these statements are false.
- $(5): \quad$ Exactly five of these statements are false.
Solution
Aiming for a contradiction, suppose more than $1$ of these statements is true.
That would mean both that:
- $\text{(a)}: \quad$ Exactly $n$ of these statements is false.
and:
- $\text{(b)}: \quad$ Exactly $m$ of these statements is false.
where $m \ne n$.
But that would mean that $\size {n - m}$ of the statements is both true and false at the same time.
That would contradict the Principle of Non-Contradiction.
Hence no more than $1$ of these statements can be true.
That would mean either $(4)$ or $(5)$ must be true.
Aiming for a contradiction, suppose $(5)$ is true.
But then that would mean that $(5)$ itself must be false.
Again, that would contradict the Principle of Non-Contradiction.
Hence it follows that $(4)$ must be true, and all the others false.
$\blacksquare$
Sources
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.3$