# 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.

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