Difference Between Interval Notations

Question

What is the difference, if any, between $x \in {(1,3)}$ and $1<x<3$ . Is it right to assume that in the second case $'x'$ might lie anywhere in the interval, but the first case tells us that all the values in the parentheses "must" be possible for a number?

Answer 1

$1 < x < 3$ and $x \in (1, 3)$ mean exactly the same thing: the open interval $(a, b)$ is by definition the set $\{x \mid a < x < b\}$ so the assertion $x \in (1, 3)$ is equivalent to the assertion $1 < x < 3$. Don't try to read too much into "elegant variations" of notation.