MATHS (R K FAI): Set theory class 11 mathematics

Set : - A well defined collection of objects is called a set. The objects of the set are called its members or elements.

Example : - i ) The collection of vowels in english alphabet.

ii) The collection of all rivers of India.

Representation of a set

There are two ways to represent a given set.

1) Roster or Tabular Form

In this form , we list all the members of the set withing curly brackets and separate these by commas.

For Example: -

i) The set A of all even numbers less than 15

A={2,4,6,8,10,12,14}

ii) The set B of vowels in English alphabet

B = { a, e, i,o,u}

2) Set builder form

In this form , we write a variable (say x ) representing any member of the set which is followed by a colon ‘:’ and there after we write the property satisfied by each member of the set and then enclose the whole description within braces.

For Example

i) The set A of all even natural numbers less than 15

A = {x : x is an even natural number less than 15}

ii) The set B = { 1, 4, 9, 16, 25 ,…….} in the builder form can be written as

B = { x : x is the square of a natural number }

Type of Sets: -

1. Empty Set - A set which does not contain any element is called the empty set or the null set or the void set. It is denoted by Φ or { }.

For Example : -

i) The collection of natural numbers less than 1

2. Singleton Set - A set that contains only one element is called a singleton set.

For example

i) {O}

ii) { x : 3x – 1 = 8}

3 Finite Set - A set that contains a limited number of different elements is called a finite set.

For Example

i) A = {a,e,i,o,u}

ii) B = { x : x is a month of year}

4) Infinite Set - A set that contains an unlimited number of different elements is called an infinite set.

For Example: -

i)The set of even natural numbers i.e. {2, 4, 6….}

ii) The set of all points on a line segment.

Cardinal Number of a Finite Set : -

The number of different elements in a finite set A is called the cardinal number of A and it is denoted by n(A) or O(A).

For Example.

i) A = {a,e,i,o,u} , then n(A) = 5

ii) A = {2, 3, 5, } then n (A) = 3

Note - The cardinal number of the empty set is zero and the cardinal number of a singleton set is one. The cardinal number of an infinite set is never defined.

Some Standard sets of number: -

1. Natural numbers - N = { 1,2,3,…………}

2. Whole numbers - W = { 0, 1, 2, 3, ……….}

3. Integers - I or Z = {………, -2, -1, 0, 1,2,……….}

4. Rational numbers - Any number which can be expressed in the form q/p where p, q ∈ I and q ≠0 is called a rational number. Thus 8/6 , -6/7 etc. are rational numbers . The set of rational number is denoted by Q.

5. Real numbers: - All rational as well as irrational numbers are real numbers. Thus – 3, 0, 5/3,√2 etc.are all real numbers. The set of real numbers is denotes by R.

6. Irrational numbers:- T ={x:x∈R and x∉Q} So, √2, √5, √11, √21 are irrational numbers .

7. Positive rational numbers: - The set of positive rational numbers is denoted by Q+

8. Positive real numbers : - The set of positive real numbers is denoted by R+

5. Equivalent sets: - Two (finite) sets A and B are called equivalent if they have same number of elements : -

For example: - A = {a,b,c,d,e} and B = {2,3,5,7,9} then n(A) = 5 = n(B)

6. Equal sets : - Two sets A and B are said to be equal if they have exactly the same elements, we write it as A = B.

For example

A = {1,2} and B = {2, 1,1,2,1} , then A = B

Subset : - Let A, B be any two sets , then a is called a subset of B if every member of A is also a member of B. We write it as A⊂ B (read as “A is a subset of B’).

If A ⊂B, then B is a supper set of A. write it as B⊃ A.

Propers Subset : - Let A be any set and B be a non – empty set, then A is called a proper subset of B if every member of A is also a member of B and ther exist atleast one element in B which is not a member of A.

If A is a proper subset of B, then A⊂B, A≠B.

Note: -

1. A⊂ A i.e. every set is a subset of itself , but not a proper subset. A subset which is not a proper subset is called an improper subset.

2. Every set has only improper subset.

3. Empty set is a proper subset of every set except itself.

4. If A is a set with n(A) = m, then the number of subset of A = (2)^m and the number of proper subset of A = (2)^m- 1.

Operations on sets: -

1 Union of two sets : - the union of two sets A and B written as A⋃B (read as ‘A union B’) is the set consisting of all the elements which belongs to A or B or both.

Thus A⋃B={x:x€Aor x€B}

For Example

If A = { 1, 2, 3, 4, 5} and B = {1,3,5, 7, 9} , then

A⋃B={1,2,3,4,5,7,9}

2 Intersection of two sets: - The intersection of two sets A and B, written as A∩B (read as ‘A intersection B’) is the set consisting of all the elements which belong to both A and B. Thus

A∩B={x:x€Aand x€B}

For Example

If A = { 1, 2, 3, 4, 5} and B = { 1,3,5,7,9} , then

A∩B={1,3,5}

3 Difference of two Sets: - Let A,B be two sets, then A – B is the set consisting of all the elements

which belongs to A but do not belong to B. Thus.

A-B={x:x∈A or x∉ B}

Similarly,

B-A={x:x∈Bor x∉A}

For Example

Let A = {1,2,3,4,5} and B = { 1, 3, 5, 7, 9}

then A –B = { 2,4} and b- A = { 7, 9}

Remark: - The sets A – B , A∩B and B-A are mutually disjoint i.e. the intersection of any two of these set is the empty

4 Complement of a set : - Let U be the universal set and A be any set then the complement of A,

denoted by A’, is the set consisting of all the elements of U which do not belong to A.

Thus.

A' =( x:x∈U and x∉A)= U - A

For Example , Let U = {1,2,3, …..,10}

and A ={ 1,2,3,5,7,9} , then A’ = {4,6,8,10}

Remark: -

i) If A is a sub set of the universal set U, then A’ is also a subset of U.

ii) (A’)’ = A

Venn diagram: - Most of the ideas about sets and the various relationship between them can be visualized by means of geometric figure known as ‘Venn diagrams’. Usually the universal set U is denoted by a rectangle and its subsets by circles within the rectangle.

Venn Diagrams in different Situations

1. SUBSET : -