sets notes for class 11

Sets theory for class 11 cbse  

Number and their representation :-

Even number       all integer numbers divided by 2                      ±2, ±4, ±6..........

Odd number         all integers number not divided by 2               ±1, ±3, ±5,............

Prime number      all integer number divided by 1 and itself only  

                               ±2, ±3, ±5, ±7, ±11  ...... (0 and 1 are not prime numbers)   

N : the set of all natural numbers      1,2,3,4,5,....................................

Z : the set of all integers                     ..........-3 ,-2 ,-1 ,0, 1, 2, 3...........

Q : the set of all rational numbers     all numbers leaving root values like √2 , √3.... etc

R : the set of real numbers                all numbers leaving negative root values like √-2 , √-3.... etc

Z+ : the set of positive integers        1,2,3,4,5 ....................................(same as natural number)

Q+ : the set of positive rational numbers all positive numbers leaving root values like √2 , √3.... etc

R+ : the set of positive real number   all positive numbers leaving root (negative vales ) like √-2 , √-3.... etc.

Set :-

A set is a well-defined collection of objects is known as set. All sets are collection of object but all collection cannot be a set.

(i) Usually sets are denoted by capital letters A, B, C, X, Y, Z, etc.

(ii) Elements or members of a set are represented by small letters a, b, c, x, y, z, etc.

Let consider as set A = {a,b,c,d}

Here a is an element of a set A, We will write a ∈ A, and read as “a is belongs to  A” .

And ‘e’ is not an element of a set A, we write b ∉ A and read as “b does not belong to A”.

Methods of representing a set:

There are two methods to represent a set

(i) Roster or tabular form

(ii) Set-builder form.

(i) Roster or tabular form

In roster form, all the elements of a set are listed in braces { } and separated by commas. Order of elements doesn’t matter. 

For example,

Set of all prime positive integers less than 9 can be written as in roaster form as {2, 3, 5, 7} or {7, 3,5,2 }

(ii) Set-builder form.

In set-builder form, all the elements of a set should have a single common property which is not possessed by any element outside the set.

For example,

 Set of all prime positive integers less than 9 can be written as inset-builder form

{x:x is a prime natural number , 0<x<9  } or {x:x is a prime natural number between 0 and 9 }.

We shell read “x:x” as “ x is such that x”.

Types of sets

Empty Set

If a set does not contain any element, it is called the empty set or the null set or the void set. The empty set is represented by the symbol φ or { }.

{φ} is not a empty set because it has a element φ.

Examples: {x : x is a number more than 5 and less than 4 }

Finite and Infinite Sets

If a set is empty or consists of a definite number of elements, it will call finite set else it will call infinite set.

Example  A= {x:x is a positive prime number less than 10} is a finite set because it will have fixed number of element {2,3,5,7}.

B = {x:x is a positive prime number} is a infinite set because it don’t  have fixed number of elements.

Equal Sets

If two sets A and B are have exactly the same elements, they will call equal set and we will write A = B. Otherwise, the sets will said to be unequal and we will write A ≠ B

Examples:

(i)                  Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.

(ii)                Let C = {1, 4, 3, 5} and D = {3, 1, 4, 2}. Then C ≠ D.

Singleton set.

If a set A has only one element, we call it a singleton set. Example A= { a } is a singleton set.

Subsets

A set A is said to be a subset of a set B if every element of A is also an element of B.

Symbolically , we will write A ⊂  B

Properties of sets

(1)          (1)                                                    A ⊂  B           if     a ∈ A                     ⇒               a ∈ B

Read above expression as “A is subset of B if a belongs to A  implies that  a belongs to B”.

(2)                                            If  A ⊂  B                and         B ⊂  A                     ⇔                   A = B, 

Read above expression as “if A is subset of B and  B is a subset of A      if and only if  a is equal to B”.

(3)  φ is a subset of every set. And every set is subset of it self.

(4) Number of subset of a set is always 2ⁿ  where n is number of elements in set.

(5) Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A. For example, A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.

Power Set

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A).

Example,  if A = { 1, 2 }, then P( A ) = { φ,{ 1 }, { 2 }, { 1,2 }}

Number of elements in power set, n [ P (A) ] = 4 = 2²

If a set has n number of elements, number of elements in power set =2ⁿ.

Universal Set

A set said to be a universal set if it contains all objects including itself. Universal set is represented by “U”.

Intervals as subsets of R

Open interval :

If a, b ∈  R and a < b, the set of real numbers { y : a < y < b} is called an open interval and is denoted by (a, b). All the points between a and b belong to the open interval (a, b) but a, b do not belong to this interval.

Infinity ( – ∞, ∞ ) always written in open interval.

Open interval

Close interval

If a, b ∈  R and a < b, the set of real numbers {x : a ≤ x ≤ b} is called closed interval and is denoted by [ a, b ]

All the points between and including a and b belong to the close interval [a, b].

Close interval

Intervals closed at one end and open at the other

[ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.

Close and open interval 

( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a.

Open and close interval 

Venn Diagrams

When the relationships between sets represented by means of diagrams, it is known

as Venn diagrams. In venn diagram rectangle is used for universal set and circles for sub set.

Union of sets  if A and B be any two sets, the union of A and B consists of all the elements of A and all the elements of B, the common elements written only once. The symbol ‘∪’ is used to denote the union. Symbolically, we write A ∪ B and read it  as ‘A union B’.

Or

The union of two sets A and B is the set C which consists of all those elements which are either in A or in B  (including those which are in both). In symbols, we write. A ∪ B = { x : x ∈A or x ∈B }

union of A and B

Example  Let A = { 1, 4, 6, 8} and B = { 6, 8, 10, 12}  Then  A ∪  B will { 1, 4, 6, 8, 10, 12}.

Intersection of sets  if A and B be any two sets, the Intersection of A and B consists of all common elements of A and all the elements of B, the common elements written only once. The symbol ‘∩’ is used to denote the Intersection. Symbolically, we write A  ∩ B and read it  as ‘A Intersection B’.

Or

The intersection of two sets A and B is the set of all those elements which belong to both A and B.

In symbols, we write. A ∩ B = { x : x ∈A and x ∈B }

 Intersection of A and B

Example  Let A = { 1, 4, 6, 8} and B = { 6, 8, 10, 12}  Then  A ∩ B will { 6, 8 }.

Disjoint sets

If A and B are two sets such that A ∩ B = φ, then A and B are called disjoint sets.

Difference of sets 

The difference of the sets A and B  is the set of elements in which element belong to A but not to B  or set of  Element of set A which don’t belongs to set B

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

Example Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }.

A – B = set of elements of set A which don’t not belongs to set B = { 1, 3, 5 }

B – A = set of elements of set B which don’t not belongs to set A =  { 8 }.

Complement of a Set

Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A. ( A′ )′ = A.

Complement of set A is represented by A′ . Thus, A′ = {x : x ∈ U and x ∉ A }. Or A′ = U – A.

De Morgan’s laws:

The complement of the union of two sets is the intersection of their complements. (A ∪ B)´ = A′ ∩ B′  and

The complement of the intersection of two sets is the union of their complements. (A ∩ B )′ = A′ ∪ B′

Some Properties of Complement Sets

1. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ

2. De Morgan’s law:  (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B )′ = A′ ∪ B′

3. Law of double complementation : (A′ )′ = A

4. Laws of empty set and universal set φ′ = U and U′ = φ.

Formulas for number of elements:-

n ( A ∪  B ) = n ( A ) + n ( B ) – n ( A ∩ B )                  ...(1)

If A ∩ B = φ, then

n ( A ∪ B ) = n ( A ) + n ( B )                                           ...(2)

n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )        ...(3)