## Note on Sets

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### Introduction

Generally, a set is denoted by the capital letters of English alphabet A, B, C etc. and their members by small letters a,b,c etc. For example, the set of English vowels may be denoted by the capital letter V whereas its members a, e, i, o, u by small letters.
Expressing it in the form of the set
V = {a, e, i, o, u}

To indicate a member belonging to a set, symbol $$\in$$ V, and is read as "a belongs to V"

To indicate that something doesn't belong to a set a symbol ∉ V, and is read as 'p doesn't belong to V'

The union of two sets is denoted by "U". For example, A U B is the union of two sets A and B. It is read as 'A union B'. This operation includes the two sets without repetition. The union of two sets A and B is A U B.

The intersection of two sets is denoted by ∩. For example, A ∩ B represents the intersection of A and B. It is read as 'A intersection B'. This operation includes the elements of the two sets belonging to both of them. If there is no any element common between two sets then A∩B = ø, where ø is an empty set. Here, the intersection set of two sets A and B is the newest set A ∩ B.

#### Description of Sets

1. Listing or rooster method: The elements are listed inside the brackets, { }. Eg: N = {1,2,3,4...}
2. Descriptive method: The common properties of elements of sets are described by words. Eg: N = {the counting numbers 1 and greater than 1}
3. Set builder or rule method: The elements are represented by a variable stating their common properties. Eg: N = {x: x ∈ N}

#### Types of Sets

Empty or Null Set: If a set contains no elements then, it is null set. It is denoted by { } or Φ. For example: A = {A set of cows with three legs} $$\therefore$$A = { }

Singleton or Unit Set: If a set contains only one elments then, it is called singleton set. For example: A = {The set of highest mountain in the world}

Finite Set: If a set contains finite number of elements i.e. countable collection then, it is called finite set. For example: A = {a set of even numbers less 20}

Infinite Set: Set containing uncountable or unlimited numbers/elments is known as infinite set. For example: A = {a set of all odd numbers}

Equal Set: Two sets having same elements are called equal sets. For example: A = {2, 4, 6, 8, 10} and B =n {the five multiples of 2} then, A = B

Equivalent Set: Two or more sets having same number of elements are called equivalent sets. It is denoted by '∼' sign. For example: If X = {1, 2, 3, 4, 5} and Y = {a, b, c, d, e}. Then n(X) = 5 and n(Y) = 5 $$\therefore$$ X and Y are equivalent sets i.e.X ∼ Y.

Universal Set: A set which contains all the subsets of a set. For example: If E = { the set of even numbers} and O = {the set of odd numbers} , we can make these sets from the set W = {whole numbers} , we can make these sets from the set W = {whole numbers}. Therefore, W is the universal set for the two sets E and O.

Subset: The set made by the elements of the universal set is a subset of that universal set . For example a universal set U = {whole numbers from 1 to 30}
i.e U = {1, 2, 3.....29, 30}

Proper Subset: Let A and B be two sets where B is the subset of A. Then B is said to be the proper subset of A if B has at least one elements less than set A. it is denoted by B ⊂ A. For example: If A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4} then, B is said to be the proper subset of set A. $$\therefore$$B ⊂ A

Overlapping Set:Two sets having some elements in common are overlapping sets. For example : A ∩ B = {6} where 6 is common to both sets. Hence, A and B are called overlapping sets.

Disjoint set:If there is no element common between two sets then the sets are called disjoint sets. For example, the set of even numbers and the set off odd numbers up to 10 are disjoint sets.

#### Use of Venn-Diagram

In the 20th century, mathematician John Venn represented the operations on sets and subsets in a simple way by means of a figure. These figures for sets are given the name Venn diagrams after his name. Venn diagrams are basically used to solve verbal problems in mathematics.

Overlapping Set:

Two sets having some elements in common are overlapping sets. In the given Venn-diagram, 4, 5, 6 are common elements in both sets A and B, then A and B are said to be overlapping sets.

Disjoint Set:

If there is no element common between two sets then the sets are called disjoint sets. In the given Venn-diagram, there are no common elements for A and B.

#### Operations of Set:

There are mainly four operations of sets. They are:-
1.) Union of set

Let U be the universal set and A and B be the subsets of U, The set of all members that belong to either setA or set B or both. A and B is the union of sets. It is denoted by A∪B.

In the given venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

$$\therefore$$ A∪B = {1, 2 , 3 ,4, 5, 6, 7, 8, 9}

The shaded region on the Venn-diagram represents A∪B.

2.) Intersection of Set

Let U be the universal set and A and B be the subsets. The set of elements belonging to both sets A and B is the intersection of these set. It is denoted by A∩B.

In the given Venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

$$\therefore$$ A∩B = {4, 5, 6}

The shaded region on the Venn-diagram represents A∩B.

3.) Complement of a set

Let U b the universal set and A be its subsets. Then the complement of set A, denoted by Ac or A' is the set of elements of U which do not belong to A.

In the given Venn-diagram

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

A = {1, 2, 3, 4, 5, 6} and

B = {4, 5, 6, 7, 8, 9} then

$$\therefore$$ A' = {7, 8, 9, 10, 11, 12}

$$\therefore$$ B' = {1, 2, 3, 10, 11, 12}

4.) Difference of Set

Let A and B be the subsets of the universal set U, then thedifferenceof two sets A and B is the set (a-) that represents the elements of A which are not in B.

$$\therefore$$ A - B = A - (A∩B)

Also, the shaded region in the given Venn-diagram represents B - A.

$$\therefore$$ B - A = B - (A∩B)

#### Cardinality of Sets

The number of elements in a set is known as a cardinal number. For example, if A = {a, b, c, d, e, f} then the cardinality of A is 6. It is written as n(A) = 6.

Here in the given Venn-diagram

U = {a, b, c, d, e, f, g, h, i, j}

n(U) = 10

A = {a, b, c, d, e, f}

n(A) = 6

B = {a, b, c, g, h}

n(B) = 5

If A and B are overlapping sets, the number of elements in non-overlapping parts A and B are denoted by no(A) and no(B) respectively.

no(A) represents the elements of A only.

no(B) represents the elements of B only.

no(A) = n(A - B) = n(A) - n(A∩B)

no(B) = n(B - A) = n(B) - n(A∩B)

In the above venn-diagram

n(A) = 6

no(A) = 3

n(B) = 5

no(B) = 2

• A set is a collection of distinct objects, considered as an object in its own right.
• Sets are one of the most fundamental concepts in mathematics.
• The elements of sets are enclosed in braces {}.
• The membership of an element of a set is denoted by ∈ {read as belong to} and non-membership is denoted by ∉ {read as does not belong to}.
.

### Very Short Questions

Here , given set H ={h , e, a ,d,s} and T = {t , a , i , l , s}
$$\therefore$$ H∩T = {h , e , a , d} ∩ {t , a , i , l , s} = {a , s}
In venn diagram ,
Here , the shaded region represents H∩T.

(i) . U = {a , c , d , e , f , g , h , i} , X = {c , f , g} , Y = {c , d , e , i , f} and Z = {b , c , i}

ii) . X∪Y∪Z = {g , f , c} ∪ {c , d , e , f , i} ∪ {c , i , b} = {b , c , d ,e , f , g , i}

(iii) . X∩Y = {g , f, c} ∩ {d , f ,c, e , i} = {c , f}

(iv) . (X ∩ Y) ∩ Z = ({c , f, g} ∩ {c , d , e , f , i}) ∩ {b , c , i} = {f , c} ∩ {b , c , i} = {c}

Here , the realtion between M and D , M∩D = Ø
Relation between K and M ; M⊂K
Relation between K and D ; D⊂K
Relation between K , M and D ; M ∪ D = K
Here , D and M are subsets of K and K is subsets of U.
K ,M and D all are subsets of U.
Now , the above relation in Venn diagram as is shown below ,

(ii) U = All students
S = School level students
C = Campus level students
G = All femlae students

Here , S , C and G all are subsets of U.
Here , the relation between S and C , S ∩ C = Ø . Again relation between S and G is S ∩ G . Here , Ø is an empty sets

Let M and C denote the set of students who drink milk and curd respectively.
Let U be the universal set.

Now , n(U)= 60 , n(M) = 30 , n(c) = 25 and n (M∩C) = 10.
The Venn diagram of above informations is shown as below

(ii) . We know that ,
n(M∪C) = n(M) + n(C) - n(M∩C) = 30 + 25 - 10 = 45.

Again , n $$\overline{M∪C}$$ = n(U) - n(M∩C) = 60 - 45 = 15

Hence, the number of students who did not drink both milk and curd = 15 Ans.

Let M and S represent the sets of studentswho passed in matematics and science respectively ,
Now , n(M) = 70 , n(S) = 70

n(M∩S) = 50
Now , given information can be represented in the alongside Venn diagram

Here , U be the universal set $$\therefore$$ n(U) = 100
we know that ,
n(M∪S) = n(M) + n(S) - n(M∩S)
= 70 + 70 - 50
= 90

Now ,
n $$\overline{M∪S}$$= n(U) - n(M∪S)
= 100 - 90 = 10.

Hence , the number who failed in both Math and Science. = 10 Ans.

Let C and V denote the set of students who like to play cricket and volleyball respectively. Let u be the universal set .

Then n (C) = 20 , n(V0 = 15n and n(C∪V) =30.
By formula , n (C∪V) = n(C) + n(V) - n(C∩V)
or , 30 = 20 + 15 - n(C∩V)
or , 30 = 35 - n(C∩V)
$$\therefore$$ n(C∩V) = 30 - 35 = 5

Therefore , number of students who like both games = 5Ans.

the above given infomation may be represented by the following Venn diagram.

Let A and b denote the set of students who like to eat apple and orange respectivele. Let U denote the universal set.

Then , n(U) = 50 , n(A) = 18 , n(B) = 15 and n $$\overline{A∪B}$$ = 10

(i). n (A∪B) = N (u) - n $$\overline{A∪B}$$ = 50 - 10 = 40
Let n (A∩B) = x
We know that ,
n (A U B) = n(A) + n(B) + n(A∩B)
or , 40 = 18 + 15 +x
or , 40 = 33 + x
$$\therefore$$ x = 40 - 33 = 7
$$\therefore$$ The number of students who like to eat a applr\e.
n(A) = n(A) + x = 18 + 7 = 25 Ans.

(ii) . The number of students who like to eat oranges.
N(B) = n(B) = x = 15 + 7 = 22 Ans.

(iii). The Venn diagram

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{2 , 7}

{2 , 4}

{1 , 4}

{1 , 7}

{k , k}
{c , k}
{c , c}
{c , i}
• ### If A = {u , v , w , x} and B = {i , o , u , w} , find (A∪B).

{a , b, ,c , d , e}

{i , s , u ,r , q , e}

{i , o , u ,v , w , x}

{i , o , u ,v , w , }

{1 , 2 , ,3}

{4 , 3}

{2 , 5 , 6}

{6 , 8 ,10}

• ### If A = {letters in 'average'} and B = {letters in 'carriage'} , find A∩B.

{a , e , g ,r}

{a , r , g}

{a ,c , b , d}

{r , f , d , e}

• ### If P = {composite numbers of one digit} and Q = {Even numbers of one digit} then  find P∩Q.

{2 , 5 , 7 , }

{6 , 5 , 0}

{1 , 3 ,  ,8 }

{4 , 6 , 8}

{2 , 5 , 9}`

{2 , 5 , 0}

{2 , 3 , 7}

{1 , 6 , 0}

• ### If A={a,b,c,d}, B={c,d,e,f} and (overline{A∪B})={g,h}, show this information in a venn-diagram and find the universal set U.

{b,c,e,d,f,e,h}
{a,b,c,d,e,f,g,h}
{b,c,e,d,f, h}
{b,c,e,d,f,a,h}
• ### If A={1,2,3,4}, B – A={6,5} and  A∩B={3,4} then find the set B and (A∪B).

{3,4,5,6}, {1,2,3,4,5,6}

{7,4,5,6}, {1,2,3,4,5,6}

{2,4,5,6}, {1,2,3,4,5,6}

{1,4,5,6}, {1,2,3,4,5,6}

{6,6}
{6,7}
{6,4}
{5,7}

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• ## You scored /17

Forum Time Replies Report
##### Aayush

In a survey it was found that 80% people like oranges, 85% liked mangoes and 75% liked both. But 333 people liked none of them. By drawing the Venn diagram,find the number of people who were in the survey.

##### Aayush

In a survey it was found that 80% people like oranges, 85% liked mangoes and 75% liked both. But 333 people liked none of them. By drawing the Venn diagram,find the number of people who were in the survey.