Set Notation

In mathematics, groups of numbers or groups of solutions are typically expressed in a collection known as a set. The members of a set are called the elements or objects of the set. The elements of a finite-sized set are each listed once inside a pair of brackets (this is called roster notation).

Examples

$\left\{a,b,c,d\right\}\text{and}\left\{1,5,3,9\right\}$

Notice that the elements of a set do not have to be listed in order. Often, a capital letter such as A, B, or C is assigned as the name of a set (as sets can contain many elements and can be cumbersome to refer to in roster notation).

In the discussion that follows, the following sets will be used:

$A=\left\{1,2,3,4\right\}$, $B=\left\{4,5,6,7\right\}$, $C=\left\{4,5\right\}$ and $D=\left\{9\right\}$

-- the statement $1\in A$ indicates that 1 is an element of the set A.

--the statement $8\notin B$ means that 8 is not an element found in the set B.

--the statement $C\subseteq B$ is read "C is a subset of B".

Another way to interpret this is that all the elements in set C are also elements of the set B. Therefore, one set is a subset of another if every element in the first set is also an element of the second set.

--the statement $C⊈A$ shows that the C is not a subset of the set A.

This means that the set C contains at least one element that is not found in set A (in this case, $5\in C$, but $5\notin A$).

--the symbol $\varnothing$ is used to indicate a set containing no elements and this set is referred to as the empty set.