# Introduction to Sets

Sets in mathematics are simply collections of distinct objects that form a group. In this step-by-step guide, you will learn more about the definition of sets.

## Step-by-step guide tosets

In mathematics, a set is a well-defined set of objects. Sets are named and displayed in capital letters. In set theory, the elements that make up a set can be anything: people, alphabets, numbers, shapes, variables, etc.

Elements of a set

The items in a set are called elements or members of a set. The elements of an array are enclosed in curly braces, separated by commas. To indicate that an element is in a set, the symbol ‘$$∈$$’ is used. If an element is not a member of a set, it is indicated by the symbol ‘$$∉$$’.

Cardinal number of a set

The cardinal number, cardinality, or order of a set represents the total number of elements in the set. Sets are defined as sets of unique elements. One of the important conditions for defining a set is that all elements of a set must be related to each other and have a common feature.

### Representation of sets

Different set symbols are used to represent collections. They differ in how the elements are indexed. The three collection symbols used to represent collections are:

Semantic form:
A semantic symbol describes an expression to indicate what the elements of a set are. For example, set $$B$$ is a list of five prime numbers.

Roster form:
The most common form used to represent sets is roster notation, where the elements of the sets are enclosed in curly brackets separated by commas. For example, Set $$B=$${$$2,4,6,8,10$$}, which is the collection of the first five even numbers. If there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element.

Set builder form:

A set builder symbol has a special rule or expression that specifically describes the common property of all elements of a set. The set builder form uses a vertical bar in its display with text that describes the character of the collection elements.

For example, $$A=$$ { $$k | k$$ is an odd number, $$k≤ 10$$}. The statement says, all the elements of set $$A$$ are odd numbers that are less than or equal to $$10$$. Sometimes a “$$:$$” is used in the place of the “$$|$$”.

### What are the types of sets?

A set has many types, such as;

• Empty set or null set: It has no element present in it. For example: $$A=$${} is a null set.
• Finite set: It has a limited number of elements. For example: $$A=$${$$2,3,4,5$$}.
• Infinite set: It has an infinite number of elements. For example: $$A=$$ {$$x: x$$ is the set of all whole numbers}.
• Equal set: Two sets that have the same members. For example: $$A=$${$$3,4,5$$} and $$B=$${$$4,5,3$$}, Set $$A =$$ Set $$B$$.
• Subsets: A set ‘$$A$$’ is called to be a subset of $$B$$ if each element of $$A$$ is also an element of $$B$$. For example: $$A=$${$$3,4$$}, $$B=$${$$1,2,3,4$$}, then $$A ⊆ B$$.
• Universal set: A set that consists of all elements of other sets present in a Venn diagram. For example: $$A=$${$$1,4$$}, $$B=$${$$4,3$$}. The universal set here will be, $$U=$${$$1, 4,3$$}.

### Sets formulas

Sets are used in algebra, statistics, and probability. There are some important set formulas are listed below. For both overlapping sets $$A$$ and $$B$$,

• $$\color{blue}{n\left(A\:U\:B\right)=\:n\left(A\right)\:+\:n\left(B\right)-\:n\left(A\:∩\:B\right)}$$
• $$\color{blue}{n\:\left(A\:∩\:B\right)=n\left(A\right)+n\left(B\right)-n\left(A\:U\:B\right)}$$
• $$\color{blue}{n\left(A\right)=n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(B\right)}$$
• $$\color{blue}{n\left(B\right)=\:n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(A\right)}$$
• $$\color{blue}{n\left(A\:-\:B\right)=n\left(A\:U\:B\right)-n\left(B\right)}$$
• $$\color{blue}{n\left(A\:-\:B\right)=\:n\left(A\right)-\:n\left(A\:∩\:B\right)}$$

For any two sets $$A$$ and $$B$$ that are disjoint,

• $$\color{blue}{n\left(A\:U\:B\right)=n\left(A\right)+\:n\left(B\right)}$$
• $$\color{blue}{A ∩ B = ∅}$$
• $$\color{blue}{n\left(A\:-\:B\right)=n\left(A\right)}$$

### Sets – Example 1:

If set $$A=$$ {$$a,b,c$$} and set $$B=$$ {$$a,b,c,l,z,r$$}, find $$A ∩ B$$.

Solution:

$$A∩B=$$ {$$a,b,c$$}

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