# Subset: a set contained in a set

Two weeks ago, we have talked about the basics of sets.  In this post, we are going to talk about subsets.

If you understand what this means, then you have a notion of a subset.

In mathematics, if $A$ and $B$ are sets, we say that $A$ is a subset of $B$, denoted by $A \subseteq B$, if all elements of $A$ are also elements of $B$. The easiest way to illustrate this is through a Venn diagram as shown on the right.  In the Venn diagram, set $A$ is within set $B$. Therefore, all elements of set $A$ are also elements of $B$.

Example 1

Let$B$ be set of all letters in the English alphabet and $A$ be the set of vowel letters. It is clear that $A \subseteq B$ since {aeiou} are elements $A$ and also are elements of $B$.

Example 2

If we let $\mathbb {Z}$ be the set of integers, and $E$ be the set of even numbers, then $E \subseteq \mathbb{Z}$.

Example 3

Let us examine the structure of quadrilaterals in the Venn diagram below.

From the diagram, it is clear that all squares are rectangles.  If we let $S$ be the set of squares, and $R$ be the set of rectangles, then $S \subseteq R$. We can also easily see from the diagram that the following statements are true: $P \subseteq Q$$H \subseteq P$, and $S \subseteq H$. Can you think of others?

Example 4

We also have  talked about the structure of the real number system.

We have discussed  and have shown that the following statements are true:

• The set of natural numbers $\mathbb{N}$ is a subset of whole numbers $\mathbb{Z^*}$.
• The set of natural numbers  $\mathbb{N}$ is a subset of the integers $\mathbb{Z}$.
• The set of integers  $\mathbb{Z}$ is  a subset of rational numbers $\mathbb{Q}$.
• The set of integers  $\mathbb{Z}$ is a subset of the rational numbers $\mathbb{R}$.
We also introduce the universal set. The universal set is the set containing all the elements considered in a given problem (or under discussion).  In the first example, our universal set is the set of all letters in the English alphabet; in the second example, the set of integers;  in the third example, all the quadrilaterals $Q$; and in the fourth example, the set of real numbers.
Image Credit: Funny Math