Table of Contents

## Set

- A set is a collection of well defined & well distinguished objects. Simply, we can say, a set is a group of objects.
- For example, a set of all the students study in BCA, a set of natural numbers – 1,2,3,4.. , a set of odd numbers – 1,3,5,7..
- All the objects of a set are also called elements of members of a set.
- These elements are placed between curly braces –
**{ }**. - The name of a set is always an Uppercase Letter. You can use lowercase letters as its elements such as
**A = { a, b, c, d }**.

## Method of Set Representation

There are mainly two types to represent a set : (1) Roster method or Listing method (2) Rule method or Set Builder form or Property method

## Types of Sets

### Finite Set

A set which contains definite number of elements is called a finite set. For example, a set of month name in a year , {January, February, March, April, May, June, July, August, September, October, November, December}, a set of vowels in English alphabets {a, e, i, o, u}, a set of even numbers less than 10 {2, 4, 6, 8} are all finite set. Sometimes the number of elements of a set are long, still it is called finite set.

### Infinite Set

A set which has no finite number of elements is called an infinite set. Let’s say a set of all natural numbers {1,2,3,4,…}, a set of positive integers which are divisible by 4 {4, 8, 12, 16,..}. In both examples, we can not say what the last element is.

### Singleton Set

A set which has one & only one element is called singleton set. For example : A={19}

### Empty Set

If a set does not have any element, then it is called empty set or null set. It denotes by **{ }** or **Φ** (phi). For example : B = { }

### Equal Sets

Two sets A & B are said to be equal set, if each element of set A is an element of set B, & each element of set B is an element of set A. We can write A = B.

Symbolically, if **(∀x) (x ∈ A => x ∈ B) & (∀x) (x ∈ B => x ∈ A)**, Then** A = B**

We can also say, if **A ⊆ B & B ⊆ A** then **A = B**

Let A = {10, 20, 30} & B = {10, 20, 30}. Each element of A is in B and each element of B is in A. So A = B

Another example, if A = {10, 20, 30} & B = {10, 20, 30, 40}. Here each element of A is in B, but one element of B (40) is not there in A. So A ≠ B.

### Equivalent Sets

If the number of elements in set A & the number of elements in set B are same, then we can say both sets are equivalent. You can also say if it is possible to match each element of set A with one element of set B, then both sets are said to be equivalent sets. It is denoted by ≅

Let’s say, A = {1, 2, 3} & B = {x, y, z}. Here both the sets have same number of elements, that is 3. So A ≅ B

### Subset

A set A is said to be a subset of set B, if each element of set A is an element of set B. Set B is called the super set.

Symbolically, **(∀x) (x ∈ A => x ∈ B)** , then **A ⊂ B**. In notation, **A ⊂ B** **if and only if,** **x ∈ A => x ∈ B**.

Let’s say, A = {a, b} & B = {a, b, c}. Here every element of set A is included in set B. So, A ⊂ B

- Remember that, every set is a subset of that set itself , that means for A,
**A ⊆ A**. - Null set is a subset of every set, that means for A,
**Φ ⊂ A**. - There are total
**2**, n ∈ N subset of every set.^{n}**If A ⊂ B & B ⊂ C, then A ⊂ C**. - For example, A = {a, b} => Number of elements = 2
^{2}= 4. So subset of Set A = { }, {a}, {b}, {a, b}

### Proper Subset

If all the elements of set A are the elements of set B, but at least one element of super set B is not an element of set A, then set A is known as proper subset of super set B.

Let’s say, A = {a, b} & B = {a, b, c}. Here every element of set A is included in set B. So, A ⊂ B

### Universal Set

The universal set in any discussion is the totality of elements under consideration as elements of set. It is denoted by **U**.

Let’s say, **U** = {1,2,3,4,5,6,7,8,9,10,11,12} , A = {3,6,9} & B = {4,8,12}. Here all the elements of A & B are in set **U**

### Power Set

The family of all the subsets of a given set is called power set. It is denoted by P().

Say, A = { 10, 20, 30 } then P(A) = { {10, 20, 30}, {10}, {20}, {30}, {10, 20}, {10, 30}, {20, 30}, { } }

### Disjoint Sets

Any two sets are said to be disjoint sets if their intersection is null set. A & B are said to disjoint if (A ∩ B) = Φ