Set Differences

Why

We consider elements of one set which are not contained in another set.

Definition

Let $A$ and $B$ denote sets. The difference between $A$ and $B$ is the set $\Set{x \in A}{x \not\in B}$. In other words, the difference between $A$ and $B$ is the set of all points of $A$ which do not belong to $B$.

It is not necessary that $B \subset A$; the difference is called proper if $A \supset B$. This terminology is from that of proper subsets.

Notation

We denote the difference between $A$ and $B$ by $A \setminus B$. Other notations used include $-$ or $\sim$.¹

Properties

The following are straightforward.²

1. The first will conflict with convenient notation for the difference of two sets of vectors. The second will conflict with convenient notation for equivalence relations . ↩︎
2. Accounts will appear in future editions. ↩︎