[Math] List the members of these sets.

Question

List the members of these sets.
a) {x | x is a real number such that x2 = 1}

What does this symbol mean | ?
the answer is (-1,1) but how do you find the answer?

Thanks

Answer 1

The notation $A=\{x|P(x)\}$ or equivalently $A=\{x:P(x)\}$ means that the set $A$ contains all $x$ for which the condition $P(x)$ is satisfied.

So your set contains all $x$, such that $x$ is a real number and $x$ satisfies the equation $x^2=1$. In other words, your set contains only the square roots of $1$, which are (as you said) $\pm1$:$$\{x|x\in \mathbb{R},x^2=1\}=\{1,-1\}.$$