# True or false? x2≠x⟹x≠1x^2\ne x\implies x\ne 1

Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:

$$x2≠x⟹x≠1x^2\ne x\implies x\ne 1$$

I immediately answered true, but for some reason, everyone (including my classmates and math teacher) is disagreeing with me. According to them, when $$x2x^2$$ is not equal to $$xx$$, $$xx$$ also can’t be $$00$$ and because $$00$$ isn’t excluded as a possible value of $$xx$$, the sentence is false. After hours, I am still unable to understand this ridiculously simple implication. I can’t believe I’m stuck with something so simple.

Why I think the logical sentence above is true:
My understanding of the implication symbol $$⟹\implies$$ is the following:
If the left part is true, then the right part must be also true. If the left part is false, then nothing is said about the right part. In the right part of this specific implication nothing is said about whether $$xx$$ can be $$00$$. Maybe $$xx$$ can’t be $$−πi-\pi i$$ too, but as I see it, it doesn’t really matter, as long as $$x≠1x \ne 1$$ holds. And it always holds when $$x2≠xx^2 \ne x$$, therefore the sentence is true.

### TL;DR:

$$x2≠x⟹x≠1x^2 \ne x \implies x \ne 1$$: Is this sentence true or false, and why?

Sorry for bothering such an amazing community with such a simple question, but I had to ask someone.

The short answer is: Yes, it is true, because the contrapositive just expresses the fact that $1^2=1$.

But in controversial discussions of these issues, it is often (but not always) a good idea to try out non-mathematical examples:

“If a nuclear bomb drops on the school building, you die.”

“Hey, but you die, too.”

“That doesn’t help you much, though, so it is still true that you die.”

“Oh no, if the supermarket is not open, I cannot buy chocolate chips cookies.”

“You cannot buy milk and bread, either!”

“Yes, but I prefer to concentrate on the major consequences.”

“If you sign this contract, you get a free pen.”

“Hey, you didn’t tell me that you get all my money.”