# Exclusive Or/Examples/One of Five Statements is True

## 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.

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.

Hence it follows that $(4)$ must be true, and all the others false.
$\blacksquare$