Note: Here defined means existence of elements and distinct means there is no need to write repeated elements.
Some examples of sets in daily life are:
Which of the following collection are sets? Justify your answer.
Q.1. A collection of allnatural numbers less than 50.
Ans. It forms a set. The set will have the following elements.
A = {1, 2, 3, 4, 5, ................... 49}
Q.2. A collection of good hockey players in America
Ans. It doesn't form a set. Because the term "Good player" is vague and it is not well defined. However, a collection of players of hockey is a set.
Q.3. A collection of all girls in our class.
Ans. It forms a set. The elements of the above set are the names of those girls in the class.
Q.4. A collection of prime numbers less than 30
Ans. It forms a set. The set will have the following elements.
A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
Q.5. A collection of ten most talented mathematics teachers.
Ans. It doesn't form a set. Because the term "Most talented" is vague and is not well defined. However, a collection of mathematics teachers is a set.
In this chapter we will have frequent interaction with some sets, so we reserve some letters for these sets as listed below:
N: The set of natural numbers
Z: The set of integers
Z^{+}: The set of all positive integers
Q: The set of all rational numbers
Q^{+}: The set of all positive rational numbers
R: The set of all real numbers
R^{+}: The set of all positive real numbers
C: The set of all complex numbers
There are two methods to represent a set:
1. Roster Form
In roster form, a set is described by listing the elements, separated by commas, within the braces.
Examples:
(i) A set of vowels of English alphabets is described by {a, e, i, o, u}
(ii) A set of whole numbers is described by {1, 2, 3, 4, 5, 6,………..}
(iii) A set even natural numbers is described by {2, 4, 6, 8, 10,……..}
2. Set Builder Form
To understand set builder form we can take the help of some examples:
As discussed above, A = {a, e, i, o, u}
In set builder form, A = {x : x is a vowel of English alphabets} or {x  x is a vowel of English alphabets}
B = {1, 2, 3, 4, 5, 6,……….}
In set builder form, B = {x : x is a natural number} or {x  x is a natural number} or {x : x ∈ N} C = {2, 4, 6, 8, 10,……..}
In set builder form
C = {x : x = 2n, n ∈ N}
Note: The set of real numbers cannot be represented by roster form. In set builder form, A = {x : x ∈ R}
1. Empty Set or Null Set: A set containing no elements is called the empty set or null set or void set.
Reading Notation:
So, it is denoted by { } or ∅.
Example: Consider the set A = {x : x < 1, x ∈ N}
There is no natural number which is less than 1.
Therefore, A = { }, or ∅.
Note: The concept of empty set plays a key role in the study of sets just like the role of the number zero in the study of the number system.
► Cardinal Number of a Set (Order of a set)
Cardinality or order of a set is defined as the number of elements that a set has. It is denoted by n(A) or c(A) or o(A) or A or #A.
Example: A = {a, b, c} then n(A) = 3 and if A = { } or ∅ then n(A) = 0.
Note: The cardinal number of a finite set is finite.
3. Infinite Set: A set is said to be an infinite set if the number of elements in the set is not finite.
Example: Let W = The set of all whole numbers.
That is, W = {0, 1, 2, 3, ......................}
The set of all whole numbers contain infinite number of elements.
Hence, W is an infinite set.
Note: The cardinal number of an infinite set is not a finite number.
Q.1. State, whether the given set is infinite or finite:
(i) {3, 5, 7, ….}
(ii) {1, 2, 3, 4}
(iii) {….., 3, 2, 1, 0, 1, 2}
(iv) {20, 30, 40, 50, ………., 200}
(v) {7, 14, 21, …………, 2401}
(vi) {0}
(vii) {∅}
(viii) {x  x is an even natural number less than 10, 000}
(ix) {All people in the world}
(x) {x  x is a prime number}
(xi) {x  x ∈ N and x > 10}
(xii) {….., 2, 1, 0}
Ans. Finite Sets: (ii), (iv), (v), (vi), (vii), (viii) & (ix)
Infinite Sets: (i), (iii), (x), (xi) & (xii)
Q.2. State, whether the following set is infinite or finite:
(i) Set of integers
(ii) {Multiples of 5}
(iii) {Fractions between 1 and 2}
(iv) {Numbers of people in India}
(v) Set of trees in the world
(vi) Set of prime numbers
(vii) Set of leaves on a tree
(viii) Set of children in all the schools of Delhi}
(ix) Set of seven natural numbers
(x) {……, 4, , 2, 0, 2, 4, 6, 8}
(xi) {12, 9, 6, 3, 0, 3, 6, ……}
(xii) {Number of points in a line segment 4 cm long}.
Ans. Finite Sets: (iv), (vii) & (viii)
Infinite Sets: (i), (ii), (iii), (v), (vi), (ix), (x), (xi) & (xii)
Q.3. State, whether the following are finite sets, or infinite sets:
(i) {2, 4, 6, 8, …….., 800}
(ii) {5, 4, 3, 2, …..}
(iii) {…., 5, 4, 3, 2,}
(iv) {Number of C.B.S.E. school in Delhi}
(v) {x : x is an integer between 60 and 60}
(vi) {No. of electrical appliances working in your house}
(vii) {x : x is a whole number greater than 20}
(viii) {x : x is a whole number less than 20}
Ans. Finite Sets: (i), (iv), (v), (vi) & (viii)
Infinite Sets: (ii), (iii) & (vii)
4. Singleton Set: A set containing only one element is called a singleton set.
Example: Consider the set A = {x : x is an integer and 1 < x < 3}
So, A = {2}. That is, A has only one element.
Hence, A is a singleton set.
Note: {0} is not null set. Because it contains one element. That is "0".
5. Equivalent Set: Two sets A and B are said to be equivalent if they have the same number of elements.
In other words, A and B are equivalent if n(A) = n(B).
Reading notation
"A and B are equivalent" is written as A ≈ B
For example:
Consider A = {1, 3, 5, 7, 9} and B = {a, e, i, o, u}
Here n(A) = n(B) = 5
Hence, A and B are equivalent sets.
6. Equal Sets: Two sets A and B are said to be equal if they contain exactly the same elements, regardless of order.
Otherwise, the sets are said to be unequal.
In other words, two sets A and B are said to be equal if
(i) Every element of A is also an element of B and
(ii) Every element of B is also an element of A.
Fig: A and B are Equal Sets
Reading notation:
Example: Consider A = {a, b, c, d} and B = {d, b, a, c}
Set A and set B contain exactly the same elements.
And also n(A) = n(B) = 4.
Hence, A and B are equal sets.
Note: If n(A) = n(B), then the two sets A and B need not be equal. Thus, equal sets are equivalent but equivalent sets need not be equal.
Q.1. State, if the given pairs of sets are equal sets or equivalent sets:
(a) {Natural numbers less than five} and {Letters of the word ‘BOAT’}
(b) {2, 4, 6, 8, 10} and {even natural numbers less than 12}
(c) {a, b, c, d} and {∆, ○, □, ∇}
(d) {Days of the week} and {Letters of the word ‘HONESTY}
(e) {1, 3, 5, 7, ……} and set of odd natural numbers.
(f) {Letters of the word ‘MEMBER’} and {Letters of the word ‘REMEMBER’}
(g) {Negative natural numbers} and {50th day of a month}
(h) {Even natural numbers} and {Odd natural numbers}
Ans.
(a) Equivalent
(b) Equal
(c) Equivalent
(d) Equivalent
(e) Equal
(f) Equal
(g) Equal
(h) Equivalent
A set A is a subset of set B if every element of A is also an element of B.
In symbol, we write a⊆ b
Reading Notation:
Read ⊆ as "X is a subset of Y" or "X is contained in Y"
Read ⊈ as "A is a not subset of B" or "A is not contained in B".
The formula for calculating the number of subsets = 2^{n}, where n = number of elements in the set.
Note:
 If A and B are any two nonempty sets such that A ⊆ B Let x be any element such that x ∈ A ⇒ x ∈ B
 Every set is a subset of itself that means A ⊆ A
 An empty set i.e ∅ is a subset of every set
Intervals as subsets of R:
Example.1
Which of the following are correct statements for the following set A:
A = {1, 2, 3, 4, 5, 6}
(a) {2, 3} ⊂ A (b) {1, 2, 3, 4, 5,6, 7} ⊃ A
(c) 8 ⊂ A (d) {3, 5, 1, 7} ⊃ A
(e) {1} ⊂ A (f) {1, 2, 3, 4} ⊂ A
(g) { } ⊃ A (h) ϕ ⊂ A
Ans. (a) Since 2 and 3 are part of set A
(b) Since all elements of set A are present in {1,2,3,4,5,6,7}
(e) Since 1 is part of set A
(f) Since 1,2,3 and 4 are part of set A
(h) Since null set is part of all sets
Subsets are classified as:
A proper subset is one that contains a few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.
Example: If set A = {2, 4, 6}, then,
Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.
Proper Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
Improper Subset: {2,4,6}
How many subsets and proper subsets does a set have?
Note: The empty set is an improper subset of itself (since it is equal to itself) but it is a proper subset of any other set.
Q1: How many elements are there in power set if
(a) A = {ϕ}
(b) B = {a, b}
(c) C = {l, m, n}
(d) D = {4, 9}
Ans.
(a) Here n(A) = 0 ,so n(P(A)) = 2^{0} = 1
(b) Here n(B)= 2, so n(P(B))= 2^{2}= 4
(c) Here n(C)= 3, so n(P(C))= 2^{3}= 8
(d) Here n(D)=2, so n(P(D))=2^{2}= 4
Q2: Find the number of proper subsets of the following.
(a) P = {x : x ∈ N, x < 5}
(b) Q = {x : x is an even prime number}
(c) R = {x : x ∈ W, x < 2}
(d) T = { }
(e) X = {0}
(f) Y = {x : x is prime, 2 < x < 10}
Ans.
(a) Following are the elements of Set P= {1,2,3,4,}, therefore n=4
So, no. of proper subsets= 2^{n}1 =2^{4}1= 15
(b) Following are the elements of Set Q={2}, therefore n=1
So, no. of proper subsets= 2^{n}1 =2^{1}1= 1
(c) Following are the elements of Set R= {0,1,2}, therefore n=3
So, no. of proper subsets= 2^{n}1= 2^{3}1= 7
(d) Following are the elements of Set R= {}, therefore n=0
So, no. of proper subsets= 2^{n}1= 2^{0}1=0
(e) Following are the elements of Set X= {0}, therefore n=1
So, no. of proper subsets= 2^{n}1= 2^{1}1=1
(f) Following are the elements of Set Y= {3,5,7}, therefore n=3
So, no. of proper subsets= 2^{n}1= 2^{3}1= 7
Q3: Write down all the subsets of
(a) {8}
(b) {p, q}
(c) {1, 3, 5}
(d) ϕ
Ans.
(a) ∅, {8}
(b) ∅, {p}, {q}, {p, q}
(c) ∅, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}
(d) ∅
Q4: Fill in the blank spaces using the symbols ⊂ or ⊄.
(a) {1, 2, 3} ______ {1, 3, 5}
(b) ϕ ______ {4, 7, 9}
(c) {x : x is rectangle in a plane} ______ {x : x is a quadrilateral in a plane}
(d) {x : x is an odd natural number} ______{x : x is an integer}
(e) {x : x is a prime number} ______ {x: x is a composite number}
(f) {5, 10, 15, 20, 25, 30} ______ {10, 20, 30, 40}
Ans.
(a) ⊄
(b) ⊂
(c) ⊂
(d) ⊂
(e) ⊄
(f) ⊄
Q5: Given set A = {a, b, c) B = {p, q, r} C = {x, y, z, m, n, t} which of following are considered as universal set for all the three sets.
(a) P = {a, b, c, p, q, x, y, m, t}
(b) Q = {ϕ}
(c) R = {a, c, q, r, b, p, t, z, m, n, x, y}
(d) S = {b, c, q, r, n, t, p, q, x, m, y, z, f, g}
Ans. (c) as it contains all the elements of sets A, B & C.
Q6: Let A be the set of letters of the word FOLLOW. Find:
(a) A
(b) n(A)
(c) Number of subsets of A
(d) Number of proper subsets of A
(e) Power set of A
Ans.
(a) {F, O, L, W}, because we only consider unique elements in a set
(b) 4, that is the no. of elements in set A
(c) Since n=4 and no. of subsets= 2^{n }
So, no. of subsets= 2^{4}=16.
(d) Since n=4 and no. of proper subsets= 2^{n}1
For set A, no. of proper subsets= 2^{4}1= 15
(e) {∅, {F}, {O}, {L}, {W}, {F, O}, {F, L}, {F, W}, {O, L}, {O, W}, {L, W}, {F, O, L}, {F, O, W}, {F, L, W}, {O, L, W}, {F, O, L, W}}
Q7: Find the power set of the following sets.
(a) A = {a, b, c}
(b) B = {0, 7}
(c) C = {0, 5, 10}
(d) D = {x}
Ans.
(a) {∅, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}
(b) {∅, {0}, {7}, {0, 7}}
(c) {∅, {0}, {5}, {10}, {0, 5}, {0, 10}, {5, 10}, {0, 5, 10}}
(d) {∅, {x}}
Fig: Venn Diagrams for Set A and B
It is done as per the following:
The orange colored patch represents the common elements {6, 8} and the quadrilateral represents
A ∪ B.
Properties of A U B
The orange colored patch represents the common elements {6, 8} as well as the A ∩ B. The intersection of 2 or more sets is the overlapped part(s) of the individual circles with the elements written in the overlapped parts.
Example:
Properties of A ∩ B
A – B = {x : x ϵ A and x ∉ B} {converse holds true for B – A}.
Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then,
A – B = {1, 3, 5} and B – A = {8}.
The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets.
It means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set.
Some important results
(iv) Complement of Sets
The complement of a set A is the set of all the elements which are the elements of the universal set but not the elements of the A. It is represented by symbol A' or A^{c}.
Mathematically, A’ = U – A
Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A. Example: Let universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and set A = {1, 3, 5, 7, 9}, then the complement of A is given as A’ = U – A = {2, 4, 6, 8, 10}
Properties Of Complement Sets
Let's try to prove this expression.
Left hand side
Right hand side
As the last image of both left hand side and the right hand side is same
Hence,this proves that (A ∪ B)’ = A’ ∩ B’.
Ans. Since, U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. Representing this with a Venn diagram we have:
Here, A is a subset of U, represented as – A ⊂ U or
U is the super set of A, represented as – U ⊃ A
If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}, then represent A – B and B – A through Venn diagrams.
A – B = {1, 2, 3}
B – A = {6, 7, 8}
Representing them in Venn diagrams:
a. AB
b. BA
When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out.
Now, we can define the following new set.
X ∪ Y = {z  z ∈ X or z ∈ Y}
(That is, z may be in X or in Y or in both X and Y)
X ∪ Y is read as "X union Y"
Now that X ∪ Y contains all the elements of X and all the elements of Y and the figure given below illustrates this.It is clear that X ⊆ X ∪ Y and also Y ⊆ X ∪ Y
Now, we can define the following new set.
X ∩ Y = {z  z ∈ X and z ∈ Y}
(That is z must be in both X and Y)
X ∩ Y is read as "X intersection Y"
Now that X ∩ Y contains only those elements which belong to both X and Y and the figure given below illustrates this.It is trivial that that X ∩ Y ⊆ X and also X ∩ Y ⊆ Y
Now, we can define the following new set.
X\Y = {z  z ∈ X but z ∉ Y}
(That is z must be in X and must not be in Y)
X\Y is read as "X difference Y"
Now that X\Y contains only elements of X which are not in Y and the figure given below illustrates this.
Some authors use A  B for A\B. We shall use the notation A \ B which is widely used in mathematics for set difference.
Now, we can define the following new set.
X Δ Y = (X\Y) ∪ (Y\X)
X Δ Y is read as "X symmetric difference Y"
Now that X Δ Y contains all elements in X∪Y which are not in X∩Y and the figure given below illustrates this.
If X ⊆ U, where U is a universal set, then U\X is called the compliment of X with respect to U.
If underlying universal set is fixed, then we denote U\X by X' and it is called compliment of X.
X' = U\X The difference set set A\B can also be viewed as the compliment of B with respect to A.
That means if x ∈ A ⇒ x ∉ A’ and x ∈ A’ ⇒ x ∉ A
It is clear that n(A ∪ B) = n(A) + n(B), if A and B are disjoint finite set.If A and B are not disjoint, then
(i) n ( A ∪ B) = n (A) + n (B)  n ( A ∩ B)
(ii) n(A ∪ B) = n(A  B) + n(B  A) + n(A ∩ B)
(iii) n (A) = n (A  B) + n ( A ∪ B)
(iv) n (B) = n (B  A ) + n ( A ∪ B)
If there three sets A,B and C then
𝑛(A ∪ B ∪ C) = 𝑛(A) + 𝑛(B) + 𝑛(C) − 𝑛(A ∩ B) − 𝑛(B ∩ C) − 𝑛(C ∩ A) + 𝑛(A ∩ B ∩ C)
Note: But students are advised to do questions having three sets with venn diagram
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1. What is a subset? 
2. What is a superset? 
3. What is a power set? 
4. What is a universal set? 
5. How are sets represented in a Venn diagram? 

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