# Sets and Types of Sets

A set is a collection of distinct objects(elements) which have common property. For example, *cat*, *elephant*, *tiger*, and *rabbit *are animals. When, these animals are considered collectively, it's called set.

## Set Notation

The members(elements) of set is separated by comma and braces **{ }** are used outside the comma separated elements.

{cat, elephant, tiger, rabbit}

For convenience, sets are denoted by a capital letter. For example,

A = {cat, elephant, tiger, rabbit}

Here, *A* is a set containing 4 elements.

### Important Points Regarding Sets

**A Set is collection of distinct members.**Wrong: Elements are not distinct A = {a, b, c, b, c, d} Right: Elements are distinct A = {a, b, c, d}

**The number of elements in a set can be both finite and infinite.**N = {2, 4, 6} N = {..., -2, -1, 0, 1, 2, ...}

**The elements of a set can be in any order.Changing the order of elements doesn't change anything.**A = {1, 2, 3} This set can also be written as: A = {2, 1, 3} A = {3, 1, 2} and so on

## Example of Sets

**Example #1: What is the set of all vowels in English alphabet?**

V = {a, e, i, o, u}

**Example #2: What is the set of integers between 2 and 9?**

I = {3, 4, 5, 6, 7, 8}

**Example #2: What is the set of prime number?**

P = {2, 3, 5, 7, 11, ... }

## Types of Sets

### Empty set

A set which do not have any element is known as empty set. It is also called Null Set, Vacuous Set or Void Set. Empty set is denoted by ϕ.

A = {} =ϕ

### Singleton set

If a set has only one element, it's known as singleton set.** **

A = { moon }

### Finite set

Set with finite number of elements is called finite set.

S = {1, 2, 3}

### Infinite set

A set with have infinite number number of elements is called infinite set.

A= { x:x is an integer } B = { 5, 10, 15, 20, 25, ... }

### Equivalent sets

Two sets are said to be equivalent sets if they have same number of elements. For Example,

A = {a, b, c, d} B = {e, f, g, h}

Here, *A* and *B* are equivalent sets because both sets have 4 elements.

### Equal Sets

Two sets are said to be equal sets if they both have exactly same elements.

A = {1, 2, 3, 4} B = {2, 4, 3, 1}

Here, *A* and *B* are equal sets because both set have same elements (order of elements doesn't matter).

### Overlapping Sets

Two sets are said to be overlapping sets if they have at least one element common.

A = {1, 2, 3, 4} B = {3, 4, 5}

Here *A* and *B* are overlapping sets because elements *3* and *4* are common in both sets.

### Disjoint Sets

Two sets are said to be disjoint sets if they don't have common element/s.

A = {1, 2, 3, 4} B = {5, 6}

Here *A* and *B* are disjoint sets because these two sets don't have common element.

### Subset

A set *P* is a subset of set *Q* if every element of set *P *is also the member of set *Q. * Simply, if set *P* is contained in set *Q*, *P* is called subset of superset *Q.* It is denoted by **P⊂Q**.

P = {1, 2, 3} Q = {1, 2, 4, 3, 9}

Here, all three elements *1, 2, and 3* of set P is also member of set *Q. *Hence, *P* is subset of *Q*.