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.

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

Here A is the set and 1, 2, 3, 4, 5 are the elements of the set. Also we can write it as \(1\epsilon A,2\epsilon A\) 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

The elements in the sets are depicted in either Roster Form or Set Builder Form.

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, \(n\epsilon N\) and \(1\leq n\leq 4\)}

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 \(A\subseteq B\) and \(A\neq B\), then A is called the proper set of B and it can be written as \(A\subset B\)

Set Properties and Laws of Set

Commutative Property :

\(A\cup B=B\cup A\)

\(A\cap B=B\cap A\)

Associative Property :

\(A\cup (B\cup C)=(A\cup B)\cup C\)

\(A\cap (B\cap C)=(A\cap B)\cap C\)

Distributive Property :

\(A\cup (B\cap C)=(A\cup B)\cap (A\cup C)\)

\(A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\)

Demorgan’s Law :

Law of union : \((A\cup B){}’=A{}’\cap B{}'\)

Law of intersection : \((A\cap B){}’=A{}’\cup B{}'\)

Complement Law :

\(A\cup A{}’=A{}’\cup A=U\)

\((A\cap A{}’)=\)Ø

Idempotent Law And Law of null and universal set :

For any finite set A

\(A\cup A=A\)

\(A\cap A=A\)

Ø\( {}’=U\)

Ø\(=U{}'\)

Set Formulas

Some of the most important set formulas are:

For any three sets A, B and C

\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)

If \(A\cap B =\)Ø, then \(n(A\cup B)=n(A)+n(B)\)

\(n(A-B)+n(A\cap B)=n(A)\)

\(n(B-A)+n(A\cap B)=n(A)\)

\(n(A-B)+n(A\cap B)+n(B-A)=n(A\cup B)\)

\(n(A\cup B\cup C)=n(A)+n(B)+n(c)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)\)

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Practise This Question

If X = {4n - 3n - 1 : n ∈ N} and Y = { 9(n-1) : n ∈ N}, then X ∪ Y is equal to