Is ( Set S contains $x$ and only $x$ then does S equals $x$ ) true?

Question

If a set $S$ contains $x$ and only $x$, then does $S$ equal $x$?

Answer 1

No. If a box contains a pencil, the box is still a box, not a pencil.

In mathematical terms, $S=\{x\}$, but $S\neq x$.

Answer 2

No. $S$ is the set containing $x$ i.e. $S=\{x\}.$

Answer 3

Just as a further comment on the other (very good) answers, $x=\{x\}$ would contradict the axiom of regularity (if you believe in that sort of thing). So the problem is not "sets vs elements" but rather regularity. In fact in the formalization of ZFC there are no words for sets or elements just an element relation $\varepsilon$, and it is understood that everything is a set, (whose elements if it has any are then in turn other sets.) Of course as the other answers point out $x$ and $\{x\}$ belong on different levels of the cumulative hierarchy.