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Explanation from a book

So after some googling I found out that the apostrophe in a set means "complement" but in this example it must mean something else right? Aside from that, I have not seen any explanations of the apostrophe in this use case.

"$x$ and $x'$ are distinct elements of $X$" so I wanna know what this $x'$ is.

I've also encountered this kind of notation in another book in a similar case. (something about "bijective")

I'm really sorry that I'm not able to formulate this question any better. I will learn how to formulate it using the tools in my future questions.

Thank you for your time and patience in explaining.

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  • $\begingroup$ Welcome to Mathematics Stack Exchange. $x'$ is an element of $X$ distinct from $x$. You could write $y$ instead of $x'$, though that would be confusing, because it's an element of $X$, not $Y$ $\endgroup$ Sep 17 '20 at 10:46
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    $\begingroup$ Note that it's not quite an apostrophe, because in most professionally typeset works an apostrophe is curved while the mark on $x'$ is straight. If I were reading a passage with $x'$ aloud, I would say "$x$ prime". The same symbol is used in at least three completely different ways, and sometimes you can only tell what it means by reading the explanation in the same text. Later in this particular text, we are told that even though $X$ is a set, $X'$ is not the complement of $X$; instead, $X'$ is just another set that is a little bit different from $X.$ $\endgroup$
    – David K
    Sep 17 '20 at 11:02
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This is just the author's way of denoting two arbitrary (distinct) elements of the set $X$. Other options could be:

"$x$ and $x^*$ are distinct elements of $X$"

"$x_1$ and $x_2$ are distinct elements of $X$"

"$x$ and $y$ are distinct elements of $X$"

"$a$ and $b$ are distinct elements of $X$"

"$\tau$ and $\tau'$ are distinct elements of $X$"

You get the idea. Any choice of two distinct symbols would work, as long as those symbols haven't been used already for something else. Beyond this, it's purely a matter of taste and style on the part of the author.

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