# Sets

Sets, in mathematics, are an organized collection of objects and can be represented in set-builder form or in roster form.

Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set.

## Sets Theory

The set theory defines the different types of sets symbols and operation performed.

### Operations on Sets

In set theory, the operations of the sets are carried when two or more sets combined to form a single set under some of the given conditions. The basic operations on sets are:

• Union of sets
• Intersection of sets
• A complement of a set
• Cartesian product of sets.
• Set difference

## Sets Formulas

Some of the most important set formulas are:

 For any three sets A, B and C n ( A ∪ B ) = n(A) + n(B) – n ( A ∩ B) If A ∩ B = ∅, then n ( A ∪ B ) – n(A) + n(B) n( A – B) + n( A ∩ B ) – n(A) n( B – A) + n( A ∩ B ) – n(A) n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B ) n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

## Representation of Sets

The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.

### Statement Form

In statement form, the well-defined descriptions of a member of a set are written and enclosed in the curly brackets.

For example, the set of even numbers less than 15.

In statement form, it can be written as {even numbers less than 15}.

### Roster Form

In Roster form, all the elements of a set are listed.

For example, the set of natural numbers less than 5.

Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….

Natural Number less than 5 = 1,2,3,4

Therefore the set is N = { 1, 2, 3, 4 }

### Set Builder Form

The general form is, A = { x : property }

For example: Write the following sets in set builder form: A={2, 4, 6, 8}

Solution:

2 = 2 x 1

4 = 2 x 2

6 = 2 x 3

8 = 2 x 4

So, the set builder form is A = {x: x=2n, n ∈ N and 1  ≤ n ≤ 4}

Also, Venn Diagrams are the simple and best way for visualized representation of sets.

## Types of Sets

• Empty Set: A set which does not contain any element is called an empty set or void set or null set. It is denoted by { } or Ø.
• Singleton Set: A set which contains a single element is called singleton set
• Finite set: A set which consists of a definite number of elements is called finite set
• Infinite set: A set which is not finite is called infinite set
• Equivalent set: If the cardinal number of the two finite sets are equal, then it is called an equivalent set. I.e, n(A) = n(B)
• Equal sets: The two sets A and B are said to be equal if they have exactly the same elements
• Subsets: A set ‘A’ is said to be a subset of B if every element of A is also an element of B. Intervals are subsets of R
• Disjoint Sets: The two sets A and B are said to be disjoint if the set does not contain any common element
• Proper set: If A ⊆ B and A ≠ B, then A is called the proper set of B and it can be written as A⊂B

### What are the Elements of a Set?

Let us take an example:

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

Since a set is usually represented by the capital letter.  Here A is the set and 1, 2, 3, 4, 5 are the elements of the set or a member of a set. The elements that are written in the set are in any order and it cannot be repeated. All the set elements are represented in small letter in case of alphabets.  Also, we can write it as 1 ∈ A, 2 ∈ A etc. The cardinal number of the set is 5. Some commonly used sets are as follows:

• N: Set of all natural numbers
• Z: Set of all integers
• Q: Set of all rational numbers
• R: Set of all real numbers
• Z+: Set of all positive integers

### Complement of Sets

The complement of any set, say P, is the set of all elements in the universal set that are not in set P. It is denoted by P’.

Properties of Complement sets:

1. P ∪ P′ = U
2. P ∩ P′ = Φ
3. Law of double complement : (P′ )′ = P
4. Laws of empty/null set(Φ) and universal set(U),  Φ′ = U and U′ = Φ.

### Properties and Laws

 Commutative Property : A∪B = B∪A A∩B = B∩A Associative Property : A ∪ ( B ∪ C) = ( A ∪ B) ∪ C A ∩ ( B ∩ C) = ( A ∩ B) ∩ C Distributive Property : A ∪ ( B  ∩ C) = ( A ∪ B)  ∩ (A ∪ C) A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C) Demorgan’s Law : Law of union           : ( A ∪ B )’ = A’ ∩ B’ Law of intersection : ( A ∩ B )’ = A’ ∪ B’ Complement Law : A ∪ A’ = A’ ∪ A =U A ∩ A’ = ∅ Idempotent Law And Law of null and universal set : For any finite set A A ∪ A = A A ∩ A = A ∅’ = U ∅ = U’

### Example of Sets

Here are few sample examples, given to represent the elements of a set.

Example 1:

Write the given statement in three methods of representation of a set:

The set of all integers that lies between -1 and 5

Solution:

The methods of representations of sets are:

Statement Form: { I is the set of integers that lies between -1 and 5}

Roster Form: I = { 0,1, 2, 3,4 }

Set-builder Form: I = { x: x ∈ I, -1 < x < 5 }

Example 2:

Find A U B and A n B and A – B.

If A = {a, b, c, d} and B = {c, d}.

Solution

A = {a, b, c, d} and B = {c, d}

A U B  = {a, b, c, d}

A n B = {c, d} and

A – B = {a, b}