Using interval notation for sets

Question

The following is from Stitz et al's Precalculus:

Example 1.1.1/2: Express the following set of numbers using interval notation:

{x|x!=3}

my solution: $(-\infty; 3)$ and $(3;\infty)$

authors' solution: $(-\infty; 3)$ or $(3;\infty)$

Why or? If $x$ can be any set of numbers in $\mathbb{R}$ except for $3$, then isn't it evident that it includes any positive and negative numbers on the number line other than $3$?

Answer 1

Because the formula : $\text {Set} =\{ x \mid \text {Form}(x) \}$ reads :

$\text {Set}$ contains all and only those $x$ that satisfy $\text {Form}(x)$.

Now, the condition $x \ne 3$ is satisfied by every number except $3$.

If we want to write the set $\{ x \mid x \ne 3 \}$ in interval notation, we have to re-write the condition accordingly.

If we write it as :

$x \in ((-\infty, 3) \text { and } (3, \infty))$,

this means that $x$ must belong to both intervals, and there is no numebr that belongs to both.

Thus, $x \ne 3$ iff :

either $x \in (-\infty, 3)$ or $x \in (3, \infty)$.