After studying the chapter Sets you will learn the concept of a set, to represent sets in roster method and set builder method; operations on sets like union, intersection, difference and symmetric difference; to use the formula for number of elements in a set and to solve simple word problems.

## 1.4.1 Introduction – Sets

One of the fundamental concepts in modern mathematics is the notion of a set. The concept of set is vital to mathematical thought and is being used in things every branch of mathematics. Understanding set theory helps us to see things in terms of systems, to represent things into sets and begin to understand logic. You can also represent sets and operations using diagrams known as Venn diagrams and apply it to solve problems more effectively.

## 1.4.2. Concept of a set – Sets:

We often deal with a collection of objects like collection of books , teachers in a school, farmers in a village, state in a country and so on.

**A well-defined collection if objects is called a set. The objects in a set are called the elements or members of the set.**

Example 1: Consider A = {3, 5 7, 11}

- 3 is a member of A, we can write 3∈A(Read as 3 belongs to A)
- 4 is not a member of A, we write 4∉A(Read as 4 does not belong to A)
- 11 is an element of A, written as 11∈A

Example 2: Consider A = {1, 3, 5, 7, 9}. Fill in the blank spaces with the appropriate symbol ∈ or ∉.

- 1___________ A;
- 4___________ A;
- 11__________A;
- 3___________A;

Solution:

- 1∈ A
- 4∉A
- 11∉A
- 3∈ A

## 1.4.3 Representation of Sets – Sets

For the three sets namely the set A of all odd numbers less than 10, the set B of all states of South India and the set C of all even natural numbers less than or equal to 10, list out all their elements.

The list of elements of A are 1, 3, 5, 7, 9.

The list of elements of B are Karnatka, Andra Pradesh, Tamilnadu and Kerala.

The list of elements of C are 2, 4, 6, 8, 10.

We use the following notation to represent this A = {1, 3, 5, 7, 9}; B = {Karnatka, Andra Pradesh, Tamil Nadu and Kerala} and C = {2, 4, 6, 8, 10}. This method of representing a set by writing all its elements is called **Roster method ***or ***Tabular method. **So, Roster method or Tabular method is a method in which we write all the elements inside a pair of brackets {}.

### 2. Set builder method or rule method:

For many sets it may not be possible;e to represent it in Roster form. For example” The set of ll natural numbers. If we try to write this in Roster form we will not be able to list out all the elements since the set is infinite. When our list cannot be completed, we use in such cases what is known as the **Set builder method or Rule method** for representing a set. Set builder method can be used in some cases where Roster method is also possible.

In this method we write a general element x and a ‘:’ meaning such that and state its property within brackets. (i) The set of all vowels in English alphabet is represented by

V = {x : x is a vowel in English alphabet}

**Example 3: List the elements of the following sets in roster form : the set of all positive integers which are multiples of 5.**

Solution: The set of positive integers which are multiples of 5 in roster form is {5, 10, 15, 20, …}

**Example 4: Write the set A = {x : x is a natural number ≤ 10} in roster form.**

Solution:

A = {x : x is a natural number ≤ 10} So, the elements of the set are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Hence the set in roster form is A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

### Subset of a set – Sets:

Let A be the set of states in India and let B be the set of all states in South India. We know B = {Karnataka, Andra Pradesh, Kerala, Tamil Nadu}. Karnataka is the element of B, but Karnataka is also a state of India. So, Karnataka is also an element of A. Similarly, Andra Pradesh, Kerala, Tamil Nadu are also state in India. Thus every element of B is an element of A. In such situation we say B is **subset **of A. So, given a set A, as set B is called subset of A if every element of B is also an element of A.

### Equality of sets – Sets:

Let A be the set of all even natural numbers less than 10 and B be the set of natural numbers less than 10 which are not odd. Then A = {2, 4, 6, 8} and B = {2, 4, 6, 8}. Note that every element of B is also an element of A and every element of A is also an element of B. In such cases A and B are called** equal sets.**

### Empty set – Sets:

Let A be the set of prime numbers less than 2. What are the elements in A? No prime number is less than 2. So no prime numbers can belong to A. This means there is no element in A.

LetB be the set of all perfect squares which are negative, we know that perfect sqyares are all non-negative. So no perfect square belong to B. This means there is no elemnet in B. Such sets are called **Empty set.**

A set containing no element is called a **null set ***or *

**a void set.**It is denoted by ∅.

**Empty set is a subset of every set. **This means **∅ ⊆ A for every set A. **

If the elements of all sets which we are considering tthe integers, then all sets under our consideration are subsets of the set of all integers. Similarly elements of all sets which we are considering are students of your school, then, all sets under our consideration re subsets of set of all students of your school. If all sets under consideration are subsets of a particular set, this particular set is called ** universal set.** We use U to denote universal set. If we are working with subsets of natural numbers, then the set of natural number N is the universal set.

** The fixed set of which the sets we are working are subsets is called universal set.**

### Power set – Sets:

Given set A, the collection of all subsets of A is called the power set of A.It is usually denoted by P(A) or 2^{A.}

**Example: If A = {1, 2, 3} then P(A) = {∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {3, 1}, {1, 2, 3}}**

### Singleton set – Sets:

A set considering of any one element is called a **singleton set.**

Examples: {1}, {a}, {(a, b)} are singleton set.

A set A is called a **finite set** if it has finite number of elements. Otherwise it is called **infinite set**. The number of element in a finite set A is denoted by n(A) or |A|. This number is called the **cardinality** of the set A

## Sets – Exercise 1.4.3 – Solutions:

**Define a set. Give examples to illustrate the difference between a collection and a set**

Solution:

**Definition: **A set is a collection of well defined objects. The objects of a set are called elements or members of the set.

**Examples: **

a) The collection set of all prime numbers between 100 and 200.

b) The collection of all planets in the universe.

c) The collection of all fair people in the city

Here (a) and (b) are examples of sets but (c) is not one cannot define fair.

**Which of the following collection are sets?**

**(a) All the students of your school. **

**(b) Members of Indian parliament. **

**(c) The colures of rainbow. **

**(d) The people of Karnataka having green ration card. **

**(e) Good teachers in a school **

**(f) Honest persons of your village. **

Solution:

(a), (b) and (c) are sets.

(d), (e) and (f) are not sets.

**Represent the following sets in roster method:**

**(a) Set of all alphabet in English language. **

**(b) Set of all odd positive integers less than 25. **

**(c) The set of all odd integers. **

**(d) The set of all rational numbers divisible by 5. **

**(e) The set of all colors in the Indian flag. **

**(f) The set of letters in the word ELEPHANT. **

**Solution:**

** **(a) A = {a, b, c……..x, y, z}

(b) Z = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23}

(c) P = {±1, ±3, ±5….}

(d) R = {5, 10, 15……}

(e) Y = {saffron, white, green}

(f) S = {E, L, P, H, A, N, T}

**Represent the following sets by using their standard notations.**

**(a) Set of natural numbers **

**(b) Set of integers **

**(c) Set of positive integers **

**(d) Set of rational numbers **

**(e) Set of real numbers **

Solution:

(a) N = {1, 2, 3……}

(b) Z = {0, ±1, ±2, ±3…….}

(c) Z + = {1, 2, 3…….}

(d) Q = {p/q, p, q z and q ≠ 0}

(e) R = (Q U Z}

**Write the following sets in set builder form:**

**(a) {1, 4, 9, 16, 25, 36} **

**(b) {2, 3, 5, 7, 11, 13, 17, 19, 23……} **

**(c) {4, 8, 12, 16, 20, 24……} **

**(d) {1, 4, 7, 10, 13, 16……} **

Solution:

(a) A = {x: /x=k^{2} for some k€N, 1 ≤ k ≤6}

(b) P = {x/x is a prime number}

(c) X = {x/x is a multiple of 4}

(d) Z = {x/x=3n – 2 when n = 1, 2, 3…..}

**State whether the set is finite or infinite:**

**(a) The set of all prime numbers. **

**(b) The set of all sand grains on this earth. **

**(c) The set of points on a line. **

**(d) The set of all school in this world. **

Solution:

(a) infinite set

(b) finite set

(c) Infinite set

(d) finite set

**Check whether the sets A and B are disjoint**

**(a) A is the set of all even positive integers. B is the set of all prime numbers. **

**(b) A = {3, 6, 9, 12, 15……} **

**B = {19, 24, 29, 34, 39……..} **

**(c) A is the set of all perfect squares; B is the set of all negative integers. **

**(d) A = {1, 2, 3} and B = {4, 5,{1, 2, 3}} **

**(e) A is the set of all hydrogen atoms in this universe; B is the set of all water molecules on earth. **

Solution:

(a) A and B are not disjoint sets since A ∩ B = {2}

(b) A and B are not disjoint since A ∩ B = {24….}

(c) A and B are disjoint.

(d) A ∩ B = {1, 2, 3}. Hence they are not disjoint.

(e) A and B is disjoint.

## 1.4.4 Operations on Sets – Sets

### Union of sets – Sets :

Suppose D is the set of all players of cricket team and E is the set of all players of hockey team, of your school. some players may be common to both the terms. School wants to be felicitate these players in a function. when all the members are called, you see that the collection consists of all players who are either in cricket team or in hockey team. this is precisely the idea of union of sets.

Suppose A and B are two sets. The union of the sets A and B is the set of all those elements which belong to B. It is denoted by AUB(read as A union B). Thus, AUB = {x |x ∈ A or x ∈ B} Example:

If A = {1, 2, 4, 5} and B = {2, 4, 6, 8, 10} then AUB = {1, 2, 4, 5, 6, 8, 10}

We can represent the union of two sets using the Venn diagram. In the adjacent figure. A = {x, z} ; b = {y, z }{ and AUB = {x , y, z} the sets A and B are represented by circles. The shaded portion represents AB.

### Intersection of Sets – Sets

Consider again the example of cricket team and hockey team of your school where three players, say Ram, John and Ismail are playing in both the teams. Suppose the school wants to felicitate players who plays for both the teams. What will be the set of all players who felicitated? It will be the set C = {Ram, John, Ismail}.Suppose we denote the set of all cricket players by A and the set of all hockey players by B. Then C is precisely the set of all players who are both in set A and set B. Thus

C = {x : x ∈A and x ∈B}

Given two sets A and B their intersection is the set of all those elements which are both in A and B. This is denoted by A∩B (read as A intersection B). Thus, A∩B = {x |x ∈A and x ∈B}.

Example : Let A = {1,2, 3, 4} and B = {2, 4, 5, 6} Find A∩B.

Solution:

Note that only common elements of A and B are {2, 4}. Hence A∩B = {2, 4}In the adjecent Venn diagram, the shaded portion represent A∩B.

Observe that A∩B = B∩A. Thus intersection is a commutative operation.

**Two sets A and B are disjoint if and only if A∩B = ∅.**

## Sets – Exercise 1.4.4

**Find union of A and B and represent it using Venn diagram.**

**(i) A = {1, 2, 3, 4, 8, 9}, B = {1, 2, 3, 5} **

**(ii) A = {1, 2, 3, 4, 5}, B = {4, 5, 7, 9} **

**(iii) A = {1,2,3}, B = {4, 5, 6} **

**(iv) A = {1, 2, 3, ,4 ,5}, B = {1, 3, 5} **

**(v) A = {a, b, c, d}, B = {b, d, e, f} **

Solution:

(i) A U B = {1, 2, 3, 4, 5, 8, 9}

(ii) A U B = {1, 2, 3, 4, 5, 7, 9}

(iii) A U B = {1, 2, 3, 4, 5, 6}

(iv) A U B = {1, 2, 3, 4, 5}

(v) A U B = {a, b, c, d, e, f}

**Find the intersection of A and B, and respect it by Venn diagram:**

**(i) A = {a, c, d, e}, B = {b, d, e, f} **

**(ii) A = {1, 2, 4, 5}, B = {2, 5, 7, 9} **

**(iii) A = {1, 3, 5, 7}, B = {2, 5, 7, 10, 12} **

**(iv) A = {1, 2, 3}, B = {5, 4, 7} **

**(v) A = {a, b, c}, B = {1, 2, 9}**

Solution:

(i) A ∩ B = {d, e}

(ii) A ∩ B = {2, 5}

(iii) A ∩ B = {5, 7}

(iv) A ∩ B = { }

**Find A**∩**B and A**∩**B when:**

**(i) A is the set of all prime numbers and B is the set of all composite natural numbers: **

**(ii) A is the set of all positive real numbers and B is the set of all negative real numbers: **

**(iii) A = N and B = Z: **

**(iv) A = {x /x **∩ **Z and x is divisible by 6} and **

**B = {x / x **∩ **Z and x is divisible by 15} **

**(v) A is the set of all points in the plane with integer coordinate and B is the set of all points with rational coordinates **

Solution:

(i) A = {2, 3, 5, 7….}

B = {1, 4, 6….}

A U B = {1, 2, 3, 4} = N

A ∩ B = { }

(ii) A = R+

B = R+

A U B = R – {10}

i.e. A U B ={set of non zero real numbers}

A U B = { }

(iii) A = N B = N

A U B = Z

A ∩ B = N

(iv) A U B = {x / x € Z and x is divisible by 6 and 15} and

A ∩ B = {x / x € Z and x is divisible by 30}

[LCM of 6 and 15 = 30]

(v) A U B = the set of all points with rational co-ordinates = B.

A ∩ B = the set of all points with rational co-ordinates = A.

**Give examples to show that**

**(i) A **U **A = A and A **∩ **A = A **

**(ii) If A **⸦ **B, then A **U **B = B and A **∩ **B =A. can you prove these statements formally? **

Solution:

(i) If A = {2 4 6 8}

Then A U A = {2, 4, 6, 8……} = A

A ∩ A = {2, 4, 6, 8…….} = A

Hence A U A = A and A ∩ A = A

(ii) A = {1, 3, 5, 7, 9……}

B = {1, 2, 3, 4, 5……}

We see that A ⸦ B

A U B = {1 2 3 4…..} = B

A ∩ B = B

A ∩ B = {1, 3, 5……} = A

A ∩ B = A

**What is A**U Φ**and A**∩ Φ**for a set A?**

**Solution:**

** **AUΦ = A ; A ∩Φ = Φ

## 1.4.5 Complement of a set – Sets

Suppose U is a set and A⊆ U. the complement of A in U is the set of all those elements of U which are not members of A. This is denoted by

A^{C} or A’. Thus,

A’ = { x : x ∈ U but x∉ A}

For any subset A of U, we have A∩A’ = ∅ and AUA’ = U.

For any set U, U’ = ∅ and ∅’ = U.

### Difference of two sets – Sets:

Given two sets A and B, we define B ⃥ A as all those elements o B which are not in A. this is read as B difference A this is also called the complement o B in A. Thus,

B ⃥ A = {x | x ∈ B and x ∉ A}

**B ⃥ A:**

B difference A

### Symmetric difference of two sets – Sets:

Let us take A = {1, 3, 5, 7} and B = {5, 7, 8}. Then A difference B = {1, 3} and B difference A = {8}

The union of (A difference B) and (B difference A) is {1, 3, 8}. It is the union of all those elements in A which are not in B and those elements in B which are not in A. It is called the symmetric difference of A and B. t is denoted by A∆B.

For any two sets A and B A∆B = (A difference B)U(B difference A)

## Sets – Exercise 1.4.5

**If A’ = {1, 2, 3, 4}, U = {1, 2, 3, 4, 5, 6, 7, 8}, find A in U and draw Venn diagram**

Solution:

** **A’ = {5, 6, 7, 8}

**If U = {x/x € 25, x€N}. A = {x/x € U, x ≤ 15} and B = {x/x € U, 0 < x ≤ 25}, list the elements of the following sets and draw Venn diagram:**

**(i) A’ in U:**

**(ii) B’ in U**

**(iii) A\B;**

**(iv) A Δ B**

Solution:

U = {1, 2, 3, 4 ……….25}

A = {1, 2, 3, 4……….15}

B = {1, 2, 3 ….25}

(i) A’ = {16, 17, 18, 19….25}

(ii) B’ = { }

(iii) A\B = { }

(iv) A Δ B = A \ B U B \ A

= { } U {16, 17, 18… 25}

= {16, 17, 18 …..25}

**Let A and B subsets of a set U. Identify the wrong statements:**

**(i) (A’)’ = A **

**(ii) A \ B = B \ A **

**(iii) A **U **A’ = U **

**(iv) A Δ B = B Δ A **

**(v) (A \ B)’ = A’ \ B’ **

Solution:

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

A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 9}

(i) (A’)’ = {2, 4, 6, 8}

(A’)’ = {1, 3, 5, 7, 9} = A

(A’)’ = A

(ii) A \ B = {1, 3, 5, 7}

B \ A = {2, 4, 6, 8}

We see that **A \ B ≠ B \ A**

(iii) A U A’ = {1, 3, 5, 7, 9} U {2, 4, 6, 8}

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

= U

A U A’ = U

(iv) A Δ B =(A \ B) U (B \ A)

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

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

B Δ A = (B \ A) U (A \ B)

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

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

A Δ B = B Δ A

(v) A \ B = {1, 3, 5, 7}

(A \ B)’ = {2, 4, 6, 8, 9}

A’ = {2, 4, 6, 8} and B’ = {1, 3, 5, 7}

A’ \ B’ = {2, 4, 6, 8}

**Hence (A \ B)’ ≠ A’ \ B’**

**Suppose U = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. A = {3, 4, 5, 6, 9}, B = {3, 7, 9, 5} and C = {6, 8, 10, 12, 7}. Write down the following sets and draw Venn diagram for each:**

**(i) A’ **

**(ii) B’ **

**(iii) C’ **

**(iv) (A’)’ **

**(v) (B’)’ **

**(iv) (C’)’ **

Solution:

(i) A’ = {7, 8, 10, 11, 12, 13}

(ii) B’ = {4, 6, 8, 10, 11, 12, 13}

(iii) C’ = {3, 4, 5, 9, 11, 13}

(iv) A’ = {7, 8, 10, 11, 12, 13}

(A’)’ = {3, 4, 5, 6, 9} = A

(v) (B’)’ = B’ = {4, 6, 8, 10, 11, 12, 13}

(B’)’ = {3, 7, 9, 5} = B

(vi) (C’)’ = {3, 4, 5, 9, 11, 13}

(C’)’ = {6, 8, 10, 12, 7} = C

**5. Suppose U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, write down the following sets and draw Venn diagram.**

**(i) A’ **

**(ii) B’ **

**(iii) A U B**

**(iv) A ∩ B (v) (A U B)’ (vi) (A ∩B)’**

**How (A UB)’ is related to A’ and B’? What relation you see between**

**(A ∩ B)’ and A’ and B’**

Solution:

(i) A’ = {5, 6, 7, 8, 9}

(ii) B’ = {1, 3, 5, 7, 9}

(iii) A U B = {1, 2, 3, 4, 6, 8}

(iv) A ∩ B = {2, 4}

(v) (A UB)’

(A U B) = {1, 2, 3, 4, 6, 8}

(A U B)’ = {5, 7, 9}

vi) (A ∩ B)’

(A ∩ B) = {2, 4}

(A ∩ B)’ = {1, 3, 5, 6, 7, 8}

We see that (A U B)’ = A’ ∩ B’

(A ∩ B)’ = A’ U B’

**6. Find (A \ B) and (B \ A) for the following sets and draw Venn diagram.**

**(i) A = {a, b, c, d, e, f, g, h} and**

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

**(ii) A = {1, 2, 3, 4, 5, 6} and**

**B = {2, 3, 5, 7, 9}**

**(iii) A = {1, 4, 9, 16, 25} and**

**B = {1, 2, 3, 4, 5, 6, 7, 8, 9}**

**(iv) A = {x | x is a prime number less than 5} and**

**B = {x | x is a square number less than 16}**

Solution:

(i) A = { a, b, c, d, e, f, g, h}

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

A \ B = {b, c, d, f, g, h}

B \ A = {i, o, u}

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

B = {2, 3, 5, 7, 9}

A \ B = {1, 4, 6}

B \ A = {7, 9}

(iii) A = {1, 4, 9, 16, 25} and

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

A \ B = {16, 25}

B \ A = {2, 3, 5, 6, 7, 8}

(iv) A = {x | x is a prime number less than 5}

= {2, 3}

B = {x | x is a square number less than 16}

= {1, 4, 9}

A \ B = {2, 3}

B \ A = {1, 4, 9}

**7. Looking at the Venn diagram list the elements of the following sets:**

**(i) A \ B**

**(ii) B \ A**

**(iii) A \ C**

**(iv) C \ A**

**(v) B \ C**

**(vi) C \ B**

Solution:

(i) A \ B = {1, 2, 7}

(ii) B \ A = {5, 6}

(iii) A \ C = {1, 2, 3}

(iv) C \ A = {6, 8, 9}

(v) B \ C = {5, 3}

(vi) C \ B = {7, 8, 9}

**8. Find A Δ B and draw Venn diagram when:**

**(i) A = {a, b, c, d} and B = {d, e, f}**

**(ii) A = {1, 2, 3, 4, 5} and B = {2, 4}**

**(iii) A ={1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}**

**(iv) A = {1, 4, 7, 8} and B = {4, 8, 6, 9}**

**(v) A = {a, b, c, d, e} and B = {1, 3, 5, 7}**

**(vi) A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}**

Ans:

(i) A = {a, b, c, d} B = {d, e, f}

A \ B = {a, b, c}

B \ A = {e, f}

A Δ B = {a, b, c, e, f}

(ii) A = {1, 2, 3, 4, 5} B = {2, 4}

A \ B = {1, 3, 5}

B \ A = { }

A Δ B = {1, 3, 5}

(iii) A ={1, 2, 3, 4, 5} ; B = {1, 2, 3, 4, 5, 6}

A \ B = {.}

B \ A = {6}

A Δ B = {6}

(iv) A = {1, 4, 7, 8}; B = {4, 8, 6, 9}

A \ B = {1, 7]

B \ A = {6, 9}

A Δ B = {1, 6, 7, 9}

(v) A = {a, b, c, d, e} and B = {1, 3, 5, 7}

A \ B = {b, d}

B \ A = {g}

A Δ B = {b, d, g}

(vi) A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7}

A \ B = {2, 4}

B \ A = {7}

A Δ B = {2, 4, 7}