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

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

*Example 1*

Let be set of all letters in the English alphabet and be the set of vowel letters. It is clear that since {*a*, *e*, *i*, *o*, *u} *are elements and also are elements of .

*Example 2*

If we let be the set of integers, and be the set of even numbers, then .

*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 be the set of squares, and be the set of rectangles, then . We can also easily see from the diagram that the following statements are true: , , and . 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 is a subset of whole numbers .
- The set of natural numbers is a subset of the integers .
- The set of integers is a subset of rational numbers .
- The set of integers is a subset of the rational numbers .

**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 ; and in the fourth example, the set of real numbers.

*Image Credit: Funny Math*