Sets and Set Notation

A set is just a collection of objects. Each object in the set is called an element, and we list the elements inside curly braces: \{1, 2, 3\} is the set whose elements are 1, 2 and 3.

A small handful of symbols carries most of the language of sets:

x \in A — “x is an element of A”.
x \notin A — “x is not an element of A”.
B \subset A — “B is a subset of A”: every element of B is also in A.
\varnothing — the empty set, the set with no elements at all.
\mathcal{E} — the universal set: everything under consideration in the current problem.

The number of elements in a set A is written |A|. So if A = \{1, 2, 3\} then |A| = 3.

Two sets can be combined into a new set. There are three core operations.

Union A \cup B — everything in A or B (or both).
Intersection A \cap B — everything in both A and B.
Complement A' — everything in the universal set that is not in A.

The language of sets in one place:

\in — element of;
\subset — subset;
\varnothing — the empty set;
A \cup B — union (or);
A \cap B — intersection (and);
A' — complement (not).

A Venn diagram pictures all of them at once.

Picturing the operations

Two sets A and B are drawn as overlapping circles inside the rectangle, which stands for the universal set \mathcal{E}. Step forward to shade the intersection, then the union.