Difference Between Interval Notations

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?

• As far as I know, they are the same. I am not familiar with an interpretation where the endpoints "'must' be possible." Commented Jan 13, 2017 at 2:41

1 Answer

$1 < x < 3$ 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.

• I mean, if I solve a problem and get $x \in {(1,3)}$ can I tick correct an option which says $x \in {(1,4)}$ ?Thanks. Commented Jan 13, 2017 at 3:14
• Well if you got $2 < x < 3$, would $1 < x < 4$ also be correct? Commented Jan 13, 2017 at 3:26
• Um, yes, but my teacher insists that while using parentheses notation we should always write the exact set. Commented Jan 13, 2017 at 3:30
• I have no idea what your teacher could mean by that. It sounds like hogwash to me. Commented Jan 13, 2017 at 3:36