Why this statement is false?

Question

I know that variance is the square of the standard deviation.

I must to answer if is true or false this statement:

If the variance is equal to standard deviation, then they are just both equal to 1.

I answered true, because if standard deviation is $1$ , its square is $1^2$, that is the same.

But the correct answer was false, so I would like to know why it is false

Answer 1

Unfortunately, you've fallen into the pit of concluding that $P \implies Q$ is the same as $Q \implies P$. The converse is not equivalent to the original statement.

Yes, it's true that if the variance is $1$ then the standard deviation equals the variance.

No, it's not true that variance being equal to standard deviation implies that the variance is $1$; after all, as pointed out in the comments, $x^2 = x$ has two solutions.

Answer 2

In logic terms, you are confusing a statement with its converse.

It's true that "If $x = 1$, then $x^{2} = x$".

But it's false that "If $x^{2} = x$, then $x = 1$.

Answer 3

The true version of that statement is:

If the variance is equal to the standard deviation $(\sigma^2 = \sigma)$, then they are both equal to one, or both equal to zero, or both infinite.