Set

A set is a collection of distinct objects, called elements or members. The elements can be numbers, symbols, points, lines, shapes, functions, physical states, or even other sets.

Sets are usually written with curly brackets, for example $A=\{1,2,3\}.$ Membership is written $x\in A$; non-membership is written $xotinA$.

Subsets

If every element of $A$ is also an element of $B$, then $A$ is a subset of $B$: $A\subseteq B.$ For example, the natural numbers are contained in the integers: $\mathbb{N}\subseteq \mathbb{Z}$. This is different from membership: $2\in \mathbb{Z}$, but $\mathbb{N}\subseteq \mathbb{Z}$.

Operations

• Union: $A\cup B$ contains elements in either set.
• Intersection: $A\cap B$ contains elements in both sets.
• Difference: $A\setminus B$ contains elements of $A$ that are not in $B$.
• Cartesian product: $A\times B$ contains ordered pairs $(a,b)$.

Sets are the background language for Number sets, intervals, relations, and mappings. In physics, a state space is a set of possible states; a trajectory is often a function from time into that set.

Common notation

Membership:

$$x\in A$$

means ”$x$ is an element of $A$.” Non-membership is

$$xotinA.$$

Subset notation:

$$A\subseteq B$$

means every element of $A$ is also in $B$. Sets are the background language for functions, probability spaces, geometry, and almost every serious maths definition.