# Set notation: How to express union of a set with a single element?

Given any simple set $S$, for example $S \subset \mathbb R$, I want to "add" a single element $x$ to $S$. It doesn't matter whether $x$ is already in $S$, so the union is the right operation to use.

I first expressed this action as follows, but then wondered if that is correct. $$x \cup S$$

Seeing that the symbol $\cup$ denotes the unit of two sets, and that $x$ is not a set but just an element, e.g. just a number, would it be better to write the following? $$\{x\} \cup S$$

Or, doesn't it matter and both notations are used?

• It would be correct to use the expression $\{ x \} \cup S$ and incorrect to use $x \cup S$. You cannot take the union of a set and something that isn't a set. – Harry Feb 26 '17 at 15:38
• The second notation is the correct onr to express what you want. $\{x\}\cup S$ will be the set consisting of those elements that are in $S$, or $x$. To the contrary, $x\cup S$ will be the set of those elements that are either elements of $S$ or elements of $x$ (don't forget that $x$ is a set as well ! -not "just an element") – Max Feb 26 '17 at 15:39
• Harry : your comment is misleading : $x$ is a set, there are no "non sets", and $x\cup S$ is not "incorrect", simply it doesn't represent what OP wants it to represent – Max Feb 26 '17 at 15:40
• @Max may be technically correct that $x \cup S$ does mean something other than $\{x\} \cup S$, but if you see an expression like $x \cup S$ in a textbook, they probably are just abusing notation/being lazy and are using it to mean $\{x\} \cup S$. – Joshua Ruiter Feb 26 '17 at 15:50

Just to summarize the comments, the proper notation would be $\{x\} \cup S,$ since $x \cup S$ implies that $x$ is just a notation for some set that could possibly contain multiple elements.