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Sets and Subsets

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The concept of sets is being used throughout the math, especially in statistics and probability theory. The study of sets is known as the "set theory". In this lesson, we are going to introduce sets and subsets which fall among the list of most important topics of mathematics.

The set theory is concerned of notion of sets and other related concepts. In this branch, we primarily deal with the things or objects, instead of numbers or expressions like other branch. In general, a set is formed when few things are collected in one place. A set can have any object or thing as its member, provided that these things should be of same type.

The completion of this chapter will enable students know the basics of set theory. They will be able to tell about definition, notion, properties and operations performed on sets. They will also be able to drawn Venn diagrams of different sets as well.

Definition

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A set is simply a collection. It is defined as a collection or group of things that have common properties. Few example of general sets that we see around are 'a set of kitchen utensils', 'a set of clothings', 'a set of types of furniture' etc.

In mathematics, we deal with the sets related to mathematical objects. Such sets could be collection of numbers, expressions, shapes and other mathematical objects. We may define a set as a well-defined group of objects. Each of these objects are known as the members or elements of the set. Two sets that have same elements are called equal sets.

A set $P$ is termed as subset of another set $Q$ if each element of $P$ is contained in $Q$.

Let's understand this with the help of a casual example. Suppose we have a set of cloths.

$C$ = $\{ pants, shirt, skirt, tie, socks \}$.

Let us take the items that exclusively belong to men in one separate set, i.e.

$M$ = $\{ pants, shirt, skirt, tie \}$, then M will be known as the subset of set $C$.

Symbolically, we write it as $M\ \subset\ C$.

Notation

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Usually, an English capital letters are used to denote a set, such as $P,\ Q,\ A,\ B,\ C$ etc. This will be the name of the set. Also, the elements of a set are represented by enclosing them in curly brackets.

For example, we may denote a set of natural numbers less than $5$ as

$A$ = $\{ 1,\ 2,\ 3,\ 4 \}$

A set of basis trigonometric ratios of a triangle may be written as :

$T$ = $\{ sin\ \theta,\ cos\ \theta,\ cosec\ \theta,\ sec\ \theta,\ cot\ \theta,\ tan\ \theta \}$

Another notation of a set is known as "set-builder notation" which describes the rule or property on the basis of which the set has been built. This is represented as under:

$A$ = $\{ x\ :\ x\ Rule \}$

where, $x$ is an element of $A$.

Let's write above examples in set notation form:

$A$ = $\{ x\ :\ x\ <\ 5,\ x\ \in\ \mathbb{N} \}$
and

$T$ = $\{ x\ :\ x$ is a trigonometric ratio$\}$

Operations

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There are 4 basic set operations which are discussed below.
1) Union

The union of two sets is defined as a set whose elements are elements of any or both the given sets. It is denoted by $\cup$. 

Let $A$ = $\{ 1,\ 2,\ 3,\ 4,\ 5,\ 6 \}$ and $B$ = $\{ 2,\ 4,\ 6,\ 8 \}$

Then, $A\ \cup\ B$ = $\{ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 8 \}$
2) Intersection

The intersection of two sets is said to be the set of all those elements which are common to both given set. It is denoted by $\cap$.

For the same above sets, $A\ \cap\ B$ = $\{ 2,\ 4,\ 6 \}$
3) Difference

The difference of a set from another is obtained by eliminating all the elements of first set from second. In above example :

$A$ - $B$ = $\{ 1,\ 3,\ 5 \}$

$B$ - $A$ = $\{ 8 \}$

Thus, in general $A$ - $B\ \neq\ B$ - $A$
4) Complement

A complement of a set $A$ is the difference of $A$ from the universal set $U$. Complement of $A$ is denoted by $\bar{A}$ or $A$'.

Let $U$ = $\{ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8 \}$, then $\bar{A}$ = $\{ 7,\ 8 \}$ and $\bar{B}$ = $\{ 1,\ 3,\ 5,\ 7 \}$

Venn Diagrams

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The Venn diagrams are pictorial representations of sets and their operations. In such diagrams, a set is usually denoted by a circle or an oval. Different operations on them are illustrated as discussed below.
1) Union

The union of two sets $A$ and $B$ is shown by whole colored portion.

Union of two sets
2) Intersection

The intersection of two sets A and B is the common portion between them shown as under.

Intersection of two sets
3) Difference

In difference, the remaining part of the set from which another is removed is highlighted.

The difference $A$ - $B$ is shown below.

Difference of Two sets A-B

While the difference $B$ - $A$ is :

Difference of Two sets B-A
4) Complement

The complement of a set A is represented by the following Venn diagram.

Complement of A or A'

Examples

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Look at the examples discussed below.
Example 1:

If $L$ = $\{ 4,\ 8,\ 12,\ 16,\ 20 \}$ and $M$ = $\{ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12 \}$, find $L\ \cup\ M,\ L\ \cap\ M,\ L$ - $M$ and $M$ - $L$.

Solution:

$L$ = $\{ 4,\ 8,\ 12,\ 16,\ 20 \}$

$M$ = $\{ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 11,\ 12 \}$

$L\ \cup\ M$ = $\{ 1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 10,\ 11,\ 12,\ 16,\ 20 \}$

$L\ \cap\ M$ = $\{ 4,\ 6,\ 8,\ 12 \}$

$L$ - $M$ = $\{ 16,\ 20 \}$

$M$ - $L$ = $\{ 0,\ 1,\ 2,\ 3,\ 5,\ 6,\ 7,\ 9,\ 10,\ 11 \}$
Example 2: 

Set $A$ = $\{ 9,\ 10,\ 11 \}$. Write all possible subsets of $A$.

Solution: 

The subsets of $A$ are

$\{  \}$

$\{ 9 \}$

$\{ 10 \}$

$\{ 11 \}$

$\{ 9,\ 10 \}$

$\{ 10,\ 11 \}$

$\{ 9,\ 11 \}$

$\{ 9,\ 10,\ 11 \}$