# Set Theory: What It Is, Types, Symbols, and Examples

Discrete Mathematics | Set Theory: In this tutorial, we will learn about the set theory, types of sets, symbols, and examples. By Prerana Jain Last updated : May 09, 2023

## What is Set Theory?

The **set** is a well-defined collection of definite objects of perception or thought and the Georg Cantor is the father of set theory. A set may also be thought of as grouping together of single objects into a whole. The objects should be distinct from each other and they should be distinguished from all those objects that do not from the set under consideration. Hence an st may be a bunch of grapes, a tea set or it may consist of geometrical points or straight lines.

A **set** is defined as an unordered collection of distinct elements of the same type where type is defined by the writer of the set.

Generally, a set is denoted by a capital symbol and the master or elements of a set are separated by an enclosed in { }.

1 E A → 1 belong to A 1 E/ A → 1 does not belong to A

## Types of Sets

The following are the types of sets in discrete mathematics:

### 1. Singleton Set

If a set contains only one element it is called to be a singleton set.

Hence the set given by **{1}, {0}, {a}** are all consisting of only one element and therefore are singleton sets.

### 2. Finite Set

A set consisting of a natural number of objects, i.e. in which number element is finite is said to be a finite set. Consider the sets

**A = { 5, 7, 9, 11} and B = { 4 , 8 , 16, 32, 64, 128}**

Obviously, **A**, **B** contain a finite number of elements, i.e. 4 objects in **A** and 6 in **B**. Thus they are finite sets.

### 3. Infinite Set

If the number of elements in a set is finite, the set is said to be an infinite set.

Thus the set of all natural number is given by **N = { 1, 2, 3, ...}** is an infinite set. Similarly the set of all rational number between ) and 1 given by

**A = {x:x E Q, 0 <x<1} is an infinite set.**

### 4. Equal Set

Two set **A** and **B** consisting of the same elements are said to be equal sets. In other words, if an element of the **set A** sets the **set A** and **B** are called equal i.e. **A = B**.

### 5. Null Set or Empty Set

A null set or an empty set is a valid set with no member.

**A = { } / phie cardinality of A is 0.**

There is two popular representation either **empty curly braces { }** or a **special symbol phie**. This **A** is a set which has null set inside it.

### 6. Subset

A subset **A** is said to be subset of **B** if every elements which belongs to **A** also belongs to **B**.

A = { 1, 2, 3} B = { 1, 2, 3, 4} A subset of B.

### 7. Proper Set

A set is said to be a proper subset of **B** if **A** is a subset of **B**, **A** is not equal to **B** or **A** is a subset of **B** but **B** contains at least one element which does not belong to **A**.

### 8. Improper Set

Set **A** is called an improper subset of **B** if and Only if **A = B**. Every set is an improper subset of itself.

### 9. Power Set

Power set of a set is defined as a set of every possible subset. If the cardinality of **A** is **n** than Cardinality of power set is **2^n** as every element has two options either to belong to a subset or not.

### 10. Universal Set

Any set which is a superset of all the sets under consideration is said to be universal set and is either denoted by **omega** or **S** or **U**.

Let A = {1, 2, 3} C = { 0, 1} then we can take S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} as universal set.

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