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.

sets-and-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

LetB 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 QH \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
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