Definition:Set Difference

Definition
The (set) difference (or difference set) between two sets $$S$$ and $$T$$ is written $$S \setminus T$$, and means the set that consists of the elements of $$S$$ which are not elements of $$T$$:
 * $$x \in S \setminus T \iff x \in S \and x \notin T$$

This can also be interpreted:
 * $$S \setminus T = \left\{{x \in S : x \notin T}\right\}$$;
 * $$S \setminus T = \left\{{x: x \in S \and x \notin T}\right\}$$.

$$S \setminus T$$ can be voiced:
 * $$S$$ slash $$T$$;
 * $$S$$ cut down by $$T$$.

Another frequently seen notation for $$S \setminus T$$ is $$S - T$$. Both notations are encountered on this website.

Some authors refer to the expression $$S \setminus T$$ as the relative complement of $$T$$ in $$S$$, but the standard definition for the latter concept requires that $$T \subseteq S$$.

Illustration by Venn Diagram
The red area in the following Venn diagram illustrates $$S \setminus T$$:


 * VennDiagramSetDifference.png

Example
For example, if $$S = \left\{{1, 2, 3}\right\}$$ and $$T = \left\{{2, 3, 4}\right\}$$, then $$S \setminus T = \left\{{1}\right\}$$, while $$T \setminus S = \left\{{4}\right\}$$.

It can immediately be seen that $$S \setminus T$$ is not commutative, in general (and in fact, that it is anticommuntative).