2
Commutative and Associative
pro perty of union and
intersection of sets.
Sets
This unit facilitates you in,
stating commutative and associative
property of union and intersection of sets.
Verification of commutative
and associative property of
union and intersection of sets.
verifying commutative and associative
property of union and intersection of sets.
Distributive property of sets.
stating and verifying distributive property
of sets.
De Morgan's law.
n(A) + n(B) = n(A B) + n(A B).
stating De Morgan's law.
verifying De Morgan's law.
stating the relationship.
n(A) + n(B) = n(A B) + n(A B)
solving problems using the relationship.
Augustus De Morgan
(1806-1871)
He was bo rn in Madurai,
Tamilnadu in India. His family
moved to England when he was 7
months old. He had his education
in Trinity college, Cambridge
England.
Demorgan's laws relate the three
basic set operations union,
intersection and complementation.
God created infinity, and man, unable to
understand infinity, created finite sets.
-Anonymous
16
UNIT-2
Let us recall the set operations which are required to learn further operations of sets.
U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {3, 5, 7, 10}
Study the following table :
Set operations
3
5
Meaning
1. Union of two sets
Venn diagram
B
7
9
10
8
A B = {1, 2, 3, 4, 5, 7,10}
It is the set of all the
elements which are
either in A or in B or in
both A and B.
U
A
2
1 23 7
1
5
10
4
4
6
2. Intersection of
two sets
It is the set of the all
elements which are
common to both A
and B.
A B = {3, 5}
U
A
B
9
6
3. Complement of sets
U is the universal set
and A U and B U
then, A| = U \ A
A| = {6, 7, 8, 9, 10}
8
2
U 11 A2 33 77B
5
4
5 10
10
6
8
4
B| = {1, 2, 4, 6, 8, 9} U
A
9
9
B
B| = U \ B
6
8
Solve the following problems on set operations and draw Venn diagrams.
1. If X = { a, b, c, d, e, f }, Y = { b, c, e, g } and Z = { c, f, p, q } find X Y, Y Z, X Z,
Y Z, X Y and X Z and represent by venn diagram.
2. If A = {x : x  N}, B = {x : x  N , and x is even}, C = {x : x  N and x is odd} and
D = {x : x  N and x is prime}, find A B, A C, A D, B C, B D, A B, A C,
A D, B C and B D.
3. If A = { 2, 4, 6, 8} and B =  find A B and A B.
4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 3, 5, 7}, B = {2, 4, 6, 8}, C = {3, 6, 9}, D = {1, 4, 9}
Write AI, BI, CI, DI and find AI BI, BI CI, CI DI, AI BI, AI DI, AI CI and CI BI.
Study the above problems on set operations.
Do these set operations exhibit any property? Are they governed by any laws?
We know the fundamental operations which are performed on numbers and the
properties they exhibit. These properties are further generalised as fundamental laws of
algebra.
Sets
17
Let us recall some of them.
Operation
Properties
Addition
Multiplication
1. Commutative property
a+b=b+a
a.b=b.a
2. Associative property
a + (b + c) = (a + b) + c
a.(b.c) = (a.b).c
3. Distributive property
a(b + c) = (ab + ac)
4. Existence of identity element
a+0=0+a=a
5. Existence of Inverse element
a + (–a) = (–a) + a = 0
a.1 = 1.a = a
a.
1
a
1
= a.a
1
Do these laws hold good for set operations also? Let us examine them.
(i) Commutative property of Union and intersection of sets:
Let us consider some examples to generalise the properties.
Study the following table.
Sets
A = {6, 7, 8}
A B
Union of sets
B A
Intersection of sets
A B
B A
{4, 6, 7, 8, 12}
{4, 6, 7, 8, 12}
{8}
{8}
{1, 2, 3, 4}
{1, 2, 3, 4}
{2, 4}
{2, 4}
{6}
{6}
B = {4, 8, 12}
A = {x : x < 5, x  N}
A = {1, 2, 3, 4}
B = {x : x is an even
number, x  N, x < 5}
B = {2, 4}
A = {x : x is a multiple
{2, 3, 4, 6, 8, 9} {2, 3, 4, 6, 8, 9}
of 2, x < 10}
A = {2, 4, 6, 8}
B = {x : x is a multiple
of 3, x < 10}
B = {3, 6, 9}
From the above table, we can state that
AB = BA and AB = BA
Union of sets is commutative and Intersection of sets is commutative
Note: Union or intersection of any number of sets can be formed by taking two sets at
a time. Union and intersection of sets are always commutative.
18
UNIT-2
(iii) Associative property of union of sets:
Consider three sets A, B and C, A = {a, b, c, d }, B = {b, c, e, f }, C = {c, e, f, g }
* Find the union of sets A and B.
A B = {a, b, c, d } {b, c, e, f } = {a, b, c, d, e, f }
* Find the union of this resultant set with set C.
(A B)C = {a, b, c, d, e, f }{c, e, f, g }
(A B)C = {a, b, c, d, e, f, g }
.....(i)
Here, first we found (A B) and later the union of sets (A B) with set C.
What is the result, if we find the union of B and C first and later its union with set A?
* Find B C, B C = {b, c, e, f }  {c, e, f, g} = {b, c, e, f, g }
* Find the union of A with (B C)
A (B C) = {a, b, c, d }  {b, c, e, f, g }
 A (B C) = {a, b, c, d, e, f, g}
......(ii)
By comparing (i) and (ii), we can observe that they are same.
 we can state that (A B)C = A (B C)
Union of sets is associative.
We can also verify the associative property of union of sets by drawing Venn diagrams.
Step 1 : Sets A, B and C
Step 2 : A B
A
A
2
B
3
4
8
4 5
6
C
6 7
8
2
4 5
3 8 6
Step
A
C
B
6
3
:
7
 B)   C
B
2
4 5
3 8 6
7
8
(A
.......(i)
C
Step 4 : B C
A
2 3
4 8
B
C
4
7
6
5
8
Step 5 : A (B  C)
A
B
2 4 5
3 8 6
......(ii)
7
C
By comparing the shaded portion of venn diagram (i) and (ii) we can state that,
(A B)C = A (B C)
(iv) Associative property of intersection of sets :
Consider three sets A, B and C, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {6, 7}
* Find the intersetction of sets A and B.
A B = {1, 2, 3, 4}  {3, 4, 5, 6} = {3, 4}
* Find the intersection of this resultant set with set C.
(A B)C = {3, 4}  {6, 7} = { } = 
....(i)
Sets
19
A
B
C
4
4
5
Here, we found (AB) first and later the intersection of set (AB) with set C.
3
6
7
What is the result, if we find the intersection of B and C first and later its intersection
with set A?
* Find B C, B C = {3, 4, 5, 6}  {6, 7} = {6}
* Find the intersection of A with (B C)
 A (B C) = {1, 2, 3, 4}  {6} = { } = 
.....(ii)
From (i) and (ii) we can observe that they are same.
So we can state that (A B)C = A (B C)
Intersection of sets is associative
Let us verify the associative properties of intersection of sets by drawing Venn
diagrams.
Step 1 : Sets A, B and C
Step 2 : (A B)
Step 3 : (A B)C
2
6
1
4
8
2
B
A
6 2 1
8 4 3
C
4 5
6 7
A
B
1
8 2
3
4
6
5 7
......(i)
C
Step 4 : B C
A
2 6
4 8
B
1
2
4
3
Step 5 : A (B C)
C
5
6
A
B
1
8 2
3
4
6
5 7
......(ii)
C
By comparing the shaded portion of Venn diagrams (i) and (ii) we state that,
(A B)C = A (B C)
(V) Distributive property of sets :
We have learnt that commutative property and associative property holds good for
operations like union and intersection of sets. Recall that only one type of set
operation, that is either union or intersection is used while stating each of the
above properties.
Is it possible to use both the operations union and intersection of sets together on
three sets? If so which property evolves in this case? Let us study about it.
Distributive property of union over intersection of sets
Consider the following three sets, K = {3, 5, 7, 9}, L = {5, 8, 9}, M = {1, 2, 3, 9}
* Find K (L M)
* Find intersection of sets L and M. L M = {5, 8, 9}  {1, 2, 3, 9} = { 9 }
* Find the union of K with (L  M)
K (L M) = {3, 5, 7, 9}  { 9 }
 K (L M)= {3, 5, 7, 9}
* Now, find the union of sets K and L and union of K and M.
K L = {3, 5, 7, 9}  {5, 8, 9} = {3, 5, 7, 8, 9}
K M = {3, 5, 7, 9}  {1, 2, 3, 9} = {1, 2, 3, 5, 7, 9}
.....(i)
20
UNIT-2
* Now find the intersection of (K L) and (K M)
(K L)  (K M) = {3, 5, 7, 8, 9}  {1, 2, 3, 5, 7, 9} = {3, 5, 7, 9}
......(ii)
From (i) and (ii) we can observe that they are same.
 We can state that, K (L M) = (K L)(K M)
 Distributive property of union over intersection holds good for sets.
Let us verify the above property by Venn diagrams. K (L M) = (K L)(K M)
LHS = K (L M)
K
L
M
LM
K
RHS = (K L)(K M)
K
L
M
K (L  M)
.....(i)
L
L
K
M
K
M
K L
L
M
(K  L)(K  M)
K M
....(i)
By comparing the shaded portion of venn diagram in (i) and (ii) we can state that,
K (L M)= (K L)(K M)
Union of sets is distributive over intersection of sets.
Distributive property of intersection over union of sets
Consider three sets K, L and M. K = {x, y, z, t}; L = {y, z};
M = {r, s, t}
* Find K (L M)
* Find union of sets L and M. L M = {y, z}  {r, s, t} = {r, s, t, y, z}
* Find the intersection of K with (L M)
K (L M) = {x, y, z, t}  {r, s, t, y, z} = {t, y, z}
.....(i)
Now, find the intersection of sets K and L, K and M and then find the union of those
two sets.
K L = {x, y, z, t}  {y, z} = {y, z} and K M = {x, y, z, t}  {r, s, t} = { t }
* Find the union of the sets K L and K  M
(K L)  (K M) = {y, z}  {t}= {t, y, z}
....(ii)
From (i) and (ii) we can observe that, the sets are same.
So, we can state that, K (L M) = (K L)(K M)
 Distributive property of intersection over union holds good for set operations.
Now, let us verify the distributive property of intersection sets over union by drawing
Venn diagram. Study the following Venn diagrams. K (L M) = (K L)(K M)
LHS = K (L M)
K
L K
M
L M
L
RHS = (K L)( K M)
K
L K
M
K (L M) ....(i)
L K
M
K L
M
KM
L
M
(K L) (K  M)
....(ii)
Sets
21
By comparing the shaded portions of (i) and (ii) we can state that,
K (L M) = (K L)(K M)
Intersection of sets is distributive over union of sets.
De Morgan's Law
So far we have learnt about the commutative, associative and distributive property
that hold good for set operations. Recall that we have also learnt to find the complementary
set of a given set.
Is there any relationship between the complement sets of given two sets?
Observe the following table. In each example two sets are given. Find their
complements and perform other operations as mentioned in each column. First one is
done for you. Can you draw any conclusion from this data? Discuss in groups.
For the set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A and B are subsets.
Sl.
No.
A
AI
B
1. {2,4,6,9}
{3,5,7,9}
2. {1,3,7,9}
{3,6,9}
3. {x:x < 5}
{x:x is even}
4. {x : x, 3  x  7}
BI
A B
{1,3,5,7,8} {1,2,4,6,8} {2,3,4,5,6,7,9}
(A B)I
AI BI
{1,8}
{1,8}
{5,7}
What can you conclude from the last two columns of the table?
We can state that in each example (A  B)I = AI  BI. This relationship was first
proposed by "Augustus De Morgan". It is called De Morgan's I Law.
Now, let us consider the example where intersection of the complement of two
sets are taken. How can De Morgan's law be stated in this case ?
Let U= {1, 2, 3, 4, 5}, A = {2, 4}, B = {3, 4, 5}
* Find AB and the complement of it. i.e. (A B) I
A B = {2, 4}  {3, 4, 5} = {4}
Complement of A B is (A B) I = U\(A B) = {1, 2, 3, 4, 5}\{4}
(A B) I = {1, 2, 3, 5} .....(i)
* Find AI and BI
A = {2, 4}  AI = {1, 3, 5}, B = {3, 4, 5}  BI = {1, 2}
* Find AI BI 
 AI BI = {1, 3, 5}  {1, 2}
 AI BI = {1, 2, 3, 5}
.....(ii)
From (i) and (ii) we can state that
(A B) I = AI B I
Note:
Verify De Morgan's II
Law for the examples
given for stating De
Morgan's I Law.
This is called De Morgan's II Law.
We can also verify De Morgan's laws by drawing Venn diagrams. Study the following
Venn diagrams.
B
A
A
BB
A
B
22
B
A
B
UNIT-2
De Morgan's (i) law : (A B) I = AI B I
A
De Morgan's (ii) law : (A B) I = AI B I
A
B
B
B
A B
A B
A
A
A
A
B
B
A
 The De Morgan's laws are (AB)I = AI B I and (AB)I = AI B I
EXERCISE 2.1
1. Verify the commutative property of union and intersection of sets for the following.
A = {l, m, n, o, p, q}
B = {m, n, o, r, s, t}
2. Given P = {a, b, c, d, e}, Q = {a, e, i, o, u} and R = {a, c, e, g}, verify associative property
of union and intersection of sets.
3. If A = {–3, –1, 0, 4, 6, 8, 10}, B = {–1, –2, 3, 4, 5, 6}, and C = {–6, –4, –2, 2, 4, 6}
show that A (B C) = (A B) (A C)
4. If U = {4, 8, 12, 16, 20, 24, 28}, A = {8, 16, 24}, B = {4, 16, 20, 28}.
Verify that i) (A B)I = AI BI and (ii) (A B)I = AI BI
5. If A = {1, 2, 3} and B = {2, 3, 4, 5} are the subsets of
U = {1, 2, 3, 4, 5, 6, 7, 8}, verify (A B)I = AI BI and (A B)I = AIBI.
6. If A = {2, 3, 5, 7, 11, 13}, B = {5, 7, 9, 11, 15} are the subsets of
U = {2, 3, 5, 7, 9, 11, 13, 15}, verify DeMorgan's laws.
7. Draw Venn diagrams to illustrate the following :
(i) (A B)
(ii) (A B)I
(iii) AI B
(v) A \ B
(vi) A B \ C)
(vii) A B C)
(x) A \ B C)
(ix) C B \ A)
(xi) (A \ B)(A \ C)
(iv) A BI
(viii) C B A)
(xii) (A B) \ A C)
8. Using Venn diagram, verify whether the following are true :
(ii) A \ B  B \ A
(i) A B = B A
I
I
I
(iii) (A B) = A B
(v) A (B C) = (A B) (A C)
(iv) A \ (B C) = (A \ B)(A \ C)
(vi) A (B C) = (A B)(A C)
9. Using Venn diagram, show that (A B)  (A \ B) = A.
Cardinality of sets
We solve many problems from daily life using mathematical ideas and sometimes using
the knowledge of sets.
Sets
23
Study the following problem :
‘‘In a certain village, among 520 members, 320 were engaged in cattle rearing, 280
members in poultry farming and 180 were doing both.’’
Suppose we want to find the number of members who are not involved in either of
the work, how to find it ? Such problems are solved using a relationship on sets.
Let us first learn it and then solve the problem.
Recall that a cardinal number is the number of distinct elements in a given set. It
is represented by 'n' of that
set.
Q
Example1: If A = {a, b, c, d}, n(A) = 4
If n(A) = 0, set A is an empty set.
Example 2 :
P
4
U
3 0 6 10
13
1 5 7 12
2
15
Here,
P = {0, 1, 3, 5, 6, 7}  n(P) = 6
Q = {6, 7, 10, 12, 13}
 n(Q) = 5
Now, let us do an activity to find the relationship between the number of elements of two
sets.
Activity : Consider the following pairs of sets.
1. A = {1, 2, 5, 9}
B = {2, 4, 6, 8, 10}
2. A = {11, 15, 17, 19}
B = {15, 20, 25}
3. A = {1, 2, 4, 5}
B={ }
For each pair of the above sets, find AB and AB and complete the table given below.
Discuss in groups. First example is done for you.
1.
n(A)
n(B)
4
5
n(A B)
8
n(A B)
1
2.
3.
Is there any relationship between n(A), n(B), n(A B) and n(A B)?
We can observe that,
The sum of n(A) and n(B) is equal to the sum of n(A B) and n(A B).
So, we can write this as n(A) + n(B) = n(A B) + n(A B)
If we know any of the three terms in this relationship, we can find the other by
rearranging the terms.
24
UNIT-2
Study the different ways of expressing the relationship
n(A B)
= n(A) + n(B)  n(A B)
n(A B)
= n(A) + n(B)  n(A B)
n(A)
= n(A B) + n(A B)  n(B)
n(B)
= n(A B) + n(A B)  n(A)
This relationship can also be proved logically.
For any two sets A and B, Prove that n(A B) = n(A) + n(B) – n(A B)
Consider the representation of sets A and B by Venn diagram.
B
A
A\B
B\A
Recall:
A\B is the difference of two
sets where x A & x B
AB
It is clear that A B is the union of A \ B, B \ A and A B.

n(A B) = n(A \ B) + n(A B) + n(B \ A)
By adding and subtracting n(AB) to RHS, we get
n(A B) = n(A \ B) + n(A B) + n(B \ A) + n(A B) – n(A B)
By grouping,
n(A B) = [n(A \ B) + n(A B)] + [n(B \ A) + n(A B)] – n(A B)
We know that, n(A \ B) + n(A B) = n(A) and n(B \ A) + n(A B) = n(B)
By substituting, we get

n(A B) = n(A) + n(B) – n(A B)
Note : If A and B are disjoint sets, then A B =  and n(A B) = 0
n(A B) = n(A) + n(B)
Now, we can conclude that,
If A and B are disjoint sets, n(A B) = n(A) + n(B)
If A and B are non-disjoint sets, n(A B) = n(A) + n(B) – n(A B)
Let us apply the formula to solve problems.
ILLUSTRATIVE EXAMPLES
Example 1: If A and B are two sets such that n(A) = 27, n(B) = 35 and n(A B) = 50,
find n(A B).
Sol: n(A) = 27, n(B) = 35, n(A B) = 50, n(A B) = ?
n(A B) = n(A) + n(B) – n(A B)
50 = 27 + 35 – n(A B)
n(A B)= 62 – 50
 n(A B) = 12
Sets
25
Example 2 : In a group of students, 75 scored first class marks in Kannada, 70 scored
first class marks in Social science and 45 scored first class in both the subjects. Find
the number of students in the group.
Sol: Number of students who scored I class in Kannada, n(K) = 75
Number of students who scored I class in Social science n(S) = 70
Number of students who scored I class in both the subjects n(K S) = 45
Number of students = n(K S) = ?
n(K S) = n(K) + n(S) – n(K S)
 n(K S) = 75 + 70 – 45 = 145 – 45 = 100.
so, there are 100 students in the group.
The above problem can also be solved using Venn diagram. Study the following Venn
diagram. It has two intersecting circles, each representing a subject. The number
0-4
5=
of students in each group is calculated as shown.
25
U
7
The result is verified as follows :
K
S
Number of students in the group
45
= 30 + 45 + 25 = 100
Example 3: Among 520 members in a village, 360 members were engaged in cattle
rearing and 280 members with poultry farming and 180 were doing both. How many
people are (i) not involved in either of the work ? (ii) engaged only in poultry farming.
Sol: Cattle rearing, n(A) = 360
Poultry farming,n(B) = 280
Both n(A B) = 180
We know, n(A B) = n(A) + n(B) – n(A B)
n(A B) = 360 + 280 – 180
n(A B) = 360 + 100 = 460
 Number of people who are engaged in the work is 460.
(i)  Number of people who are not engaged in either of the work is = 520 – 460 = 60.
 60 members of the village are not involved in either of the work.
(ii) Number of people engaged only in poultry farming = n(B) \ n (A B)
= 280 – 180 = 100
we can represent the problem by Venn diagram.
EXERCISE 2.2
I. Solve the following problems and verify the data in each case by drawing Venn
diagrams.
1. If A and B are the sets such that n(A) = 37, n(B) = 26 and n(A B) = 51, find n(A B).
2. In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like
(i) either tea or coffee (ii) neither tea nor coffee
3. In a group of passengers, 100 know Kannada, 50 know English and 25 know both.
If passengers know either Kannada or English. How many passengers are in the group?
4. In a class, 50 students offered Mathematics, 42 offered Biology and 24 offered both
the subjects. Find the number of students, who offer i) Mathematics only (ii) Biology
only. Also find the total number of students in the class.
26
UNIT-2
5. In a medical examination of 150 people, it was found that 90 had eye problem, 50
had heart problem and 30 had both complaints. What percentage of people had either
eye problem or heart problem ?
II. Solve by drawing Venn diagrams only.
1. A radio station surveyed 190 audience to determine the types of music they liked.
The survey revealed that 114 liked rock music, 50 liked folk music, 41 liked classical
music, 14 liked rock music and folk music, 15 liked rock music and classical music,
and 11 liked classical music and folk music. 5 liked all the three types of music.
Find
(i) how many did not like any of the 3 types ?
(ii) how many liked any two types only ?
(iii) how many liked folk music but not rock music ?
2. An advertising agency finds that, of 170 people, 115 use television, 110 use radio
and 130 use magazines. Also, 85 use television and magazines, 75 use television
and radio, 95 use radio and magazines, 70 use all the three.
Find, (i) how many use only television ? (ii)how many use only radio ?
(iii) how many use television and magazine but not radio ?
3. In a village, out of 120 farmers, 93 farmers have grown vegetables, 63 have grown
flowers, 45 have grown sugarcane, 45 farmers have grown vegetables and flowers, 24
farmers have grown flowers and sugarcane, 27 farmers have grown vegetables and
sugarcane. Find how many farmers have grown vegetables, flowers and sugarcane.
4. In a town 85% use bicycle, 40% use motorbike and 20% use car. Also, 32% use
bicycle and motorbike, 13% use bicycle and car and 10% use motorbike and car.
Find the percentage of people who use all the three types of vehicles.
Sets
Properties of
operation on sets
Commutative
Property
Union of sets
A B = B A
De Morgan's
Laws
Associative
Property
Intersection
of sets AB
= BA
Union of sets
A (B C) = (A B)C
Disjoint sets
n(AB) =
n(A) + n(B)
I Law
(A B) I = A I B I
Non-disjoint sets
n(AB) =
n(A)+n(B)–n(AB)
II Law
(A B) I = A I B I
Intersection of sets
A (B C) = (A B)C
ANSWERS
EXERCISE 2.2
I. 1] 12
Distributive
Property
Cardinality of sets
2] (i) 39
(ii) 11
3] 125
4] (i) 26
(ii) 18
(iii) 68
5] 73.33%