Sets

Students get a detailed knowledge on law of sets and some of the set properties from class 11 onwards. The purpose of set maths is to group a collection of related objects. It is important everywhere in mathematics because it helps for building more complex mathematical structure.

Definition

Set is represented as a collection of well defined objects 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.

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

Set Representation

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.

Solution:

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}

Types of Sets

  • Empty Set : A set which does not contain any element is called 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 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 as 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
  • Subset : 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

Set Operations

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 four basic operations on sets are:

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

Set Properties and Laws of Set

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’

Set 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)

Sample Example

A sample example is given to represent the elements of a set.

Question:

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 }

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Practise This Question

If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B ={2, 4, ......, 18} and N, the set of natural numbers is the universal set, then (A[(AB)B]) is