1. Class XI: Maths
Chapter 1: Sets
Key Formulae
1. Union of sets AB ={x:xA or xB }
2. Intersection of sets AB ={x:xA and xB }
3. Complement of a set A’ = {x: xU and xA},
A’ = U-A
4. Difference of sets A-B = {x: xA, xB} and B
–A = {x: xB, xA}
5. Properties of the Operation of Union.
a. Commutative Law:
A B = B A
b. Associative Law:
(AB) C = A (BC)
c. Law of Identity
A = A
d. Idempotent law
A A = A
e. Law of U
U A = U
6. Properties of Operation of Intersection
i) Commutative Law:
A B = B A
ii) Associative Law:
(AB) C = A (BC)
iii) Law of and U
A =, U A = U
iv) Idempotent law
A A = A
v) Distributive law
A (B C) = (A B) (A C)

2. 7. Properties of complement of sets:
a. Complement laws:
i. A A’ = U
ii. A A’ = 
b. De-Morgan’s law:
i. (A B)’ = A’ B’
ii. (A B)’ = A’ B’
c. Law of double complementation:
(A’)’ = A
d. Laws of empty set and universal set:
’ = U and U’ = 
8. Counting Theorems
a. If A and B are finite sets, and A B = then
number of elements
in the union of two sets
n(AUB) = n(A) + n(B)
b. If A and B are finite sets, A B = then
n(AU B ) = n(A) + n(B) - n(A ∩B)
c. n(A B) = n(A – B) + n(B – A) + n(A B)
d. n(A B  C) = n(A) + n(B) + n(C) – n(B∩C) –
n(A∩B) – n(A∩C) +
n(A∩B∩C)
9. Number of elements in the power set of a set
with n elements =2n.
Number of Proper subsets in the power set = 2n-2
Question: Are the following pair of sets equal? Give
reasons.
(i) A = {2, 3}; B = {x: x is solution of x2 + 5x + 6 = 0}

3. (ii) A = {x: x is a letter in the word FOLLOW}; B = {y: y is
a letter in the word WOLF}
Answer (i) no [B={-2,-3}]
(ii) yes.
Question: Let A= {1, 2, {3, 4,}, 5}. Which of the
following statements are incorrect and why?
(i) {3, 4}⊂ A
(ii) {3, 4}}∈ A
(iii) {{3, 4}}⊂ A
(iv) 1∈ A
(v) 1⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) Φ ∈ A
(x) Φ ⊂ A
(xi) ,Φ- ⊂ A
Answer F,T,T,T,F,F,F,F,F,T,F
What is a Set?
A set is a collection of discrete data items. The members of
the set can be numbers or

4. names.
Describing a Set
There are two distinct ways of describing the members of a
set. One is to list them
explicitly, like you would find in a database of names.
A = { Mark, Angela, Frank, Laura }
A couple features of sets is that order doesn’t matter, and
duplicates don’t really count.
{ Mark, Angela, Frank, Laura } = { Laura, Frank, Mark, Angela
}
and
{ Mark, Angela, Frank, Laura } = { Mark, Laura, Angela,
Mark, Frank, Laura }
Another way to define a set is to describe a mathematical
relationship.
A = {x | 2x + 6 = 0 }
The vertical bar can be read as “such that”, so that the
entire statement would be read as
“set A consists of members solving for x, such that 2 times x
plus 6 equals 0”
This same set can be listed explicitly.
A = { -3 }
x ∈ A (member / element of)
Similarly, we use a slightly different symbol to state that the
content of a variable is not a
member of a particular set.
x ∉ A (not a member / element of)
This notation is good for individual members, but what if
we are trying to compare a

5. group of set members? For that we have “subsets”. A
subset is any set whose members
are members of another set.
A = { Mark, Angela }
B = { Mark, Angela, Frank, Laura }
Set A is a subset of set B because all members of set A are in
set B. A symbol that is
commonly used is ⊆. Thus, we could write
A ⊆ B (subset)
We make one additional distinction between sets, and that
has to do with whether every
member is accounted for. If every member is accounted for,
the sets are equal. If they are
not, we have a proper subset. A proper subset is denoted
using a slightly different
symbol.
A ⊂ B (proper subset)
Thus, if two sets are the same, then one cannot be a
proper subset of the other.
A power set is a collection (set) of sets which represents
every valid subset of a set. The
symbol for the power set is a stylized P, or P. Thus, where
we have a set…
B = { Fred, Mary, Jane }
The members of the power set for set B would be
∅, {Fred}, {Mary}, {Jane}, {Fred, Mary},
{Fred, Jane}, {Mary, Jane}, {Fred, Mary, Jane}
We could also write
P (B) = { ∅, {Fred}, {Mary}, {Jane}, {Fred, Mary},

6. {Fred, Jane}, {Mary, Jane}, {Fred, Mary, Jane} }
Notice that in the case above the number of elements in set
B was 3. The number of
elements in the power set of B is 8.
The operation A – B removes those
members in set B that are in set A. If a member in set B isn’t
in set A, then nothing is
done. (There is no sense of “negative data”, so you cannot
remove what isn’t there.) In
our case we would get this:
A = { Mary, Mark, Fred, Angela, Frank, Laura }
B = { Fred, Mary, Frank, Jane }
A – B = { Mark, Angela, Laura }
Example Consider the sets
φ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol⊂ or⊄ between each of the following pair
of sets:
(i)φ . . . B
(ii) A . . . B
(iii) A . . . C
(iv) B . . . C
Solution (i)φ ⊂ B as φ is a subset of every set.
(ii)A⊄ B as 3∈ A and 3∉ B
(iii) A⊂ C as 1, 3∈ A also belongs to C
(iv) B⊂ C as each element of B is also an element of C.

7. Example Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a
subset of B ? No.
(Why?). Is B a subset of A? No. (Why?)
Example L e t A, B and C be three sets. If A∈ B and B⊂ C, is
it true that
A⊂ C? If not, give an example.
Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here
A∈ B as A = {1}
and B⊂ C. But A⊄ C as 1∈ A and 1∉ C.
Note that an element of a set can never be a subset of itself.
Subsets of set of real numbers
There are many important subsets of R. We give below the
names of some of these subsets.
The set of natural numbers N = {1, 2, 3, 4, 5, . . .}
The set of integers
Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}
The set of rational numbers Q = {x :x = p/ q, p, q∈ Z and q ≠
0}
Question Let A ={1,2}, B={1,2,3,4}, c ={5,6}, D={5,6,7,8},
Verify that (a) AX(B∩C) = (AXB)∩(AXC) (b) AXC is a subset
of BXD.
Answer take ordered pair and check the above results.
Question: Show that the following four conditions are
equivalent:
(i) A ⊂ B (ii) A – B = Φ
(iii) A ∪ B = B (iv) A ∩ B = A

8. Answer: (i)⬄(ii)⬄(iii)⬄(iv) as all elements of A are in B.
Question: Is it true that for any sets A and B, P (A) ∪ P (B) =
P (A ∪ B)? Justify your answer.
Answer by an example A= {a},B= {b},above result is not
true.
Question: For any two sets A and B prove that
P(A)UP(B)⊂P(AUB) but , P(AUB) is not necessarily a subset
of P(A)UP(B).
Answer Let X∈ P(A)UP(B)⇨ X⊂A or X⊂B ⇨ X ⊂AUB, for
other part let A={1,2} and B={3,4,5}
Then , we find X={1,2,3,4}⊂AUB, but X∉P(A),X∉P(B).SO
X∉P(A)UP(B).
Question: Using properties of sets show that
(i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A
Answer (i) (AUA)∩(AUB)=A∩(AUB)=A
(ii) (A∩A)U(A∩B)=AU(A∩B)=A
Question: Show that for any sets A and B,
A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)
Answer (A ∩ B) ∪ (A – B)= (A ∩ B) ∪ (A ∩ B’)=A∩X=A, A ∪ (B
– A)=AU(B∩A’)=(AUB)∩(AUA’)

9. = (AUB)∩X=AUB.
Question: If P(A)=P(B), show that A=B.
Sol: Let x∈A ⇨ X∈P(A) ⇨X∈P(B) ⇨X⊂B⇨x∈B⇨A⊂B
,similarily B⊂ A⇨A=B.
Question: Let A,B and C be the three sets such that
AUB=AUC and A∩B=A∩C.Show that B=C.
Answer ( AUB)∩C=(AUC )∩C
(A∩C)U(B∩C) = (A∩C)U(C∩C)
=C
Again, (AUB)∩B=(AUC)∩B
(A∩B)U(B∩B) = (A∩B)U(C∩B)
B = (A∩B)U(C∩B) ⇨B=C.
A B

10. Union The union of two or more sets can be shown in a
diagram as shown here. A’ A U

A B =Notice that the contents of both circles are shaded.
A∩B= A – B= A∩B’
C C
AUBUC=

11. Example : A market research group conducted a survey of
1000 consumers and reported that 720 consumers like
product A and 450 consumers like product B. What is the
least number that must have liked both products?
Solution n(A ∩B)= 1170 – n(AUB), n(AUB)≤ n(U) then n(A
∩B)≥170(least value) and maximum value of n(A ∩B) is
1000.
Example: A college awarded 38 medals in football, 15 in
basket ball and 20 in cricket. If
These medals went to a total of 58 men and three men got
medals in all three sports, how
Many received medals in exactly two of three sports?
Solution: n(F  = n(F) + n(B) + n(C) – n(B∩C) – n(F∩B) –
B C)
n(F∩C) +n(F∩B∩C)] or by venn diagram.
58 = 38+ 15 +20 - [ n(B∩C) + n(F∩B) + n(F∩C)
]+3
18 = [ n(B∩C) + n(F∩B) + n(F∩C) ]
Number of men who received medals in exactly two of the
three sports
= n( F∩B∩C’)+( F∩B’∩C) + (F’∩B∩C) = n(B∩C) + n(F∩B) +
n(F∩C) - 3 n(F∩B∩C)=9.
∵ n( F∩B∩C’) = n(F∩B) - n(F∩B∩C).

12. Question: In a survey of 60 people, it was found that 25
people read newspaper H, 26 read newspaper T, 26 read
newspaper I, 9 read both H and I,11 read both H and T, 8
read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the
newspapers.
(ii) the number of people who read exactly one newspaper.
Answer (i) 52,(ii) 30 [n(A  = n(A) + n(B) + n(C) –
B C)
n(B∩C) – n(A∩B) – n(A∩C) +
n(A∩B∩C)] or by venn diagram.
Question: In a survey it was found that 21 people liked
product A, 26 liked product B and 29 liked product C. If 14
people liked products A and B, 12 people liked products C
and A, 14 people liked products B and C and 8 liked all the
three products. Find how many liked product C only,
Product A and C but not product B , atleast one of three
products.
Answer n(A’∩B’∩C’)=11, n(A∩B’∩C)=4 , n(AUBUC)=54.
.
**Question: Prove that for non-empty sets
(AUBUC)∩(A∩B’∩C’)’∩C’ = B∩C’.
Answer: L.H.S.⇨ (AUBUC)∩(A’UBUC)’∩C’ =
(A∩A’)U(BUC)∩C’= (B∩C’)U(C∩C’)=R.H.S ∅
.
**Question: Let A = {(x,y):y=ex ,x∈R} and B = {(x,y):y=e-x
,x∈R}. Is A∩B empty?

13. If not find the ordered pair belonging to A∩B.
Answer: ex = e-x ⇨ e2x =1⇨ x=0, for x=0,y=1⇨ A and B
meet on (0,1) and A∩B=∅.
**Question: A and B are sets such that n(A-B)= 14+x, n(B-
A)= 3x and n(A∩B)=x,draw venn diagram to illustrate the
information and if n(A)=n(B), find x and n(AUB).
Answer: n(A)=n(B) ⇨ n(A-B)+n(A∩B)=n(B-A)+n(A∩B)⇨
x=7
n(AUB)= n(A-B)+n(A∩B)+n(B-A)=49.
**Question: If A ={1}, find number of elements in
P[P{P(A)}].
Answer: 16.
**Question: Suppose A1,A2,A3....,A30 are thirty sets each
having 5 elements and B1,B2,B3,....Bn
Are n sets each with 3 elements, let =
=S and each element of S belongs to exactly 10 of
the Ai’s and exactly 9 of the Bj’s.Then n is equal to .....
Answer: no. Of distinct elements in S= =
=15= = =45.
**Question: Two finite sets have m and n elements. The
number of subsets of the first
Set is 112 more than that of the second set. The
values of m and n are, resp.(find)
Answer: 2m-2n =112⇨ 2n(2(m-n) – 1)= 24(23 – 1).
**Question: If X={8n – 7n – 1,n∈N} and Y = {49n – 49,n∈N}.
Then find the relation b/w X,Y[X⊂ Y, Y⊂ X, X= Y, X ∩
Y=∅]
Answer: X=Y.[X= (1+7)n - 7n – 1,by binomial
expansion⇨ 49(C(n,2)+C(n,3)7+.....+C(n,n)7(n-2)) = 49(n-1)]

14. ASSIGNMENT(SETS)
Question: 1 If U = {1,2,3......,10} , A = {1,2,3,5}, B = {2,4,6,7},
then find (A-B)’.
[Answer is {2,4,6,7...10}
Question: 2 In an examination, 80% students passed in
Mathematics,72% passed in science and 13% failed in both
the subjects, if 312 students passed in both the
subjects.Find the total number of students who appeared in
the examination.
[Answer number of students failed in both the subjects =
n(M’∩S’)=13% of x=0.13x
n(U) – n,(MUS)’- = 1.52x – 312 ⇨x=480.]
Question: 3 If U = ,x :x ≤ 10, x∈ N}, A = {x :x ∈ N, x is
prime}, B = {x : x∈ N, x is even}
Write A ∩B’ in roster form.
[ Answer is {3,5,7}]
Question: 4 In a survey of 5000 people in a town, 2250
were listed as reading English Newspaper, 1750 as reading
Hindi Newspaper and 875 were listed as reading both Hindi
as well as English. Find how many People do not read Hindi
or English Newspaper. Find how many people read only
English Newspaper.

15. Answer: People do not read Hindi or English Newspaper
n*(EUH)’+ = n(U) – n(EUH) = 1875, people read only English
Newspaper n(E’∩H) = n(E) – n(E∩ H) = 1375.
Question 5 The Cartesian product AXA has 9 elements
among which are found (-1,0) and (0,1).
Find the set A and the remaining elements of AXA.
Answer (-1,0) and (0,1)∈AXA ⇨ A = {-1,0,1} and AXA = {(-1,-
1) ,( -1,0) , (-1,1) , (0,-1),(0,0) ,( 0,1) ,
( 1,-1) , (1,0),(1,1)}
Question 6 A and B are two sets such that
n(A-B) = 14 + x, n(B-A) = 3x and n(A ∩B) =x.
Draw the venn diagram to illustrate information and if
n(A) = n(B) then find the value of x.
Answer n(A-B) = 14 + x= n(A ∩B’) = n(A) - n(A ∩B)⇨n(A)
14+2x , n(B) = 4x ⇨ x=7
U
A aaaaaaaaaaaaaaa B
A-B B-A
n(A∩B)

16. Question: 7 Let A and B be two sets , prove that:
(A – B)UB = A iff B⊂A
[Hint: (A ∩B’) U B=A ⇨ (AUB) ∩U =A⇨ B⊂ A
If B⊂ A ,(A ∩B’) U B = (AUB) ∩U=A.]
Question:8 In a survey of 100 students , the number of
students studying the various languages were found to be:
English only 18,English but not Hindi 23,English and Sanskrit
8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language
24.Find:
(i) How many students were studying Hind?
(ii) How many students were studying English and
Hindi?
[Hint: answer (i) 18 (ii) 3 , use venn diagram]
Question: 9 In a survey of 500 television viewers produced
the following informations; 285 watch football, 195 watch
hockey, 115 watch basketball, 45 watch football and
basketball, 70 watch football and hockey, 50 watch hockey
and basketball, 50 do not watch any of three games. How
many watch all the three games? How many watch exactly
one of the three games?
[Hint: answer 20 ,325]
Question: 10 (i) Write roster form of {x: and 1≤ n ≤3 ,
n∈ N}

17. (ii) Write set-builder form of {-4,-3,-2,-
1,0,1,2,3,4}
[ answer { ½,2/5,3/10} , {x: x∈Z , x2 <20}
Question:11 If set A = {x:x=1/y, where y∈N},then which
of the following belongs to A:
0, 1, 2, 2/3. [1]
Question:12 If n(A) = 3, n(B) = 6 and the number of
elements in AUB and in A∩B.
Answer: A ⊆ B ⇨n(AUB)=n(B), n(A∩B)=n(A).