Sets Class 11 Notes Mathematics Chapter 1 - CBSE

Chapter : 1

What Are Sets ?

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Set

A set is a well defined collection of objects. The following points may be noted:

Objects, elements and members of a set are synonyms terms.
Sets are usually denoted by capital letters A, B, C, X, Y, Z etc.
The elements of set are represented by small letters a, b, c etc.

Types of representing a set

Tabular or Roster Form: All the elements of a set are listed, the elements are being separated by commas and are enclosed with in brackets { }. E.g. : A = {1, 2, 3, ……}
Set Builder Form: All the elements of a set posses a single common property which is not possessed by any element outside the set. E.g. : N = { x : x is a natural number}

Types of Sets

Empty or Null Set: A set which does not contain any element, is called an empty set or null set. The empty set denoted by the symbol “f ”.
Singleton Set: A set which contains only one element, is called as singleton set. e.g. : A = {2}, B = {x : x is a natural number, x – 5 = 0}
Finite and Infinite Sets: A set which is empty or consists of a definite number of elements, is called finite set otherwise, the set is called infinite. e.g. : A = {a, e, i, o, u} is a finite set and B = {x : x is an integer} is an infinite set.
Equal Sets: Two sets A and B are said to be equal, if they have exactly the same elements and we write it as A = B and read as “A is equal to B”.

Subsets

The set B said to be a subset of a set A if every element of B is also an element of A. It is represented as B ⊆ A read as “B is a subset of A”. e.g. : Let A = {1, 3, 5} and B = {x : x is a natural number}

∴ A ⊆ B

Note: If A ⊂ B and A ≠ B, then A is called a proper subset of B, and B is called super set of A.

E.g. : Q is proper subset of R.

Universal Set

If a set is subset of a given set then set is called as universal set.

E.g. : A = set of all student in your school, B = set of all science student in your class. Here A is a universal set.

Power Set

The collection of all subsets of a set is called the power set. E.g. : If A = {a, b} then subsets of A are f, {a}, {b}, {a, b}. Therefore, power set P(A) = {f, {a},{b}, {a, b}}

Note: In general, if A is set with n(A) = m then it can be shown that n{(PA)} = 2ⁿ

Operations On Sets

Union of sets

The union of two sets is the set which consists of all those elements which are either in sets (including those which are in both). The union of two sets can be represented as A ∪ B.

A ∪ B = {x : x ∈ A or x ∈ B}

Properties of operation of union

A ∪ Φ = A (Identity law)
A ∪ B = B ∪ A (Commutative law)
A ∪ A = A (Idempotent law)
(A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law)
U ∪ A = U (Universal law)

Intersection of sets

The intersection of sets is the set of all elements which are common to both sets.

A ∩ B = {x : x ∈ A and x ∈ B}

Properties of intersection of set

A ∩ A = A (Idempotent law)
(A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law)
A ∩ A = A (Idempotent law)
(A ∩ B) ∩ C = A ∩(B ∩ C) (Associative law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩C) (Distributive law)

Difference of sets

The difference of two sets A and B in this order is the set of all elements which belongs to A but not to B.

Symbolically, we write A – B and read as “A minus B”.

Thus A – B = {x : x ∈ A and x ∈ B}

Note: A – B ≠ B – A

Complement of a set

Complement set is obtained by removing the element of given set from the element of given set from the universal set, it is denoted by A.

E.g. : U = {1, 2, 3, 4, 5, 6, 7, 8, 9,10} and A = {2, 4, 6} then A' = {1, 3, 5, 7, 8, 9, 10}

Properties of Complement of a set

U' = {x ∈ U : x ∉ U} = Φ
Φ' = {x ∈ U : x ∈ Φ} = U
(A')' = A
A ∪ A' = U
A ∩ A' = Φ
(A∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'

Representation Of Initial Operations On Sets

By Venn Diagrams

Representation of sets by a big rectangle and other sets by circles inside this if any elements are common from two sets then represent two circle by intersecting them each other.

Here rectangle denotes universal set whose other element are subset. Set A and B are indicating by intersecting circles, i.e. some elements are common in both sets and C is at different place, Hence, no element of C lies in set A or B.