2.2 Algebra of Sets

There are operations that we can do on sets to form new sets. These operations can be interpreted as set-theoretic implementations of the Boolean operations and, or, & not. They also have satisfying parallels with operations and relations that we are familiar with for numbers. For the next three definitions, let $A$ and $B$ be sets.

Definition 2.6.

The union of $A$ and $B$ , $A\cup B$ , is defined as

\{x:x\in A\ \text{or}\ x\in B\}.

Where ‘or’ is inclusive, so the union includes all elements of both $A$ and $B$ .

Definition 2.7.

The intersection of $A$ and $B$ , $A\cap B$ , is defined as

\{x:x\in A\ \text{and}\ x\in B\}.

So the intersection includes only elements that are in both sets. $A$ and $B$ are said to be disjoint if and only if $A\cap B=\emptyset$ , i.e. they have no elements in common.

Definition 2.8.

The difference of $A$ and $B$ , $A\setminus B$ , is defined as

\{x:x\in A\ \text{and}\ x\notin B\}.

We can also use this operation to define the complement of a set. For example, if $A$ is a subset of some set $X$ , then the complement of $A$ in $X$ is given by $X\setminus A$ , i.e. the elements of $X$ that are not in $A$ .

Definition 2.9.

The Cartesian product of $A$ and $B$ , $A\times B$ , is defined as

\{(a,b):a\in A\ \text{and}\ b\in B\}.

So in plain speech, the Cartesian product of two sets is the set of all ordered pairs where the first component is a member of the first set and the second component is a member of the second set. It can be seen as a generalisation of the notion of Cartesian coordinates in the plane. Sometimes the Cartesian product of a set $X$ with itself is noted as $X^{2}$ instead of $X\times X$ , for example $\mathbb{Z}^{2}=\mathbb{Z}\times\mathbb{Z}$ denoting the set of all integer points in the plane.