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After studying this chapter, you will be able to:
(cid:1) understand the concept of set theory;
(cid:1) appreciate the basics of functions and relations;
(cid:1) understand the types of functions and relations; and
(cid:1) solve problems relating to sets, functions and relations.
In our mathematical language, everything in this universe , whether living or non-living, is
called an object.
If we consider a collection of objects given in such a way that it is possible to tell beyond doubt
whether a given object is in the collection under consideration or not, then such a collection of
objects is called a well-defined collection of objects.
77777.....11111 SSSSSEEEEETTTTTSSSSS
A set is defined to be a collection of well-defined distinct objects. This collection may be listed
or described. Each object is called an element of the set. We usually denote sets by capital
letters and their elements by small letters.
EEEEExxxxxaaaaammmmmpppppllllleeeee:::::A = { a, e, i, o, u}
B = {2, 4, 6, 8, 10}
C = {pqr, prq, qrp, rqp, qpr, rpq}
D = {1, 3, 5, 7, 9}
E = {1,2}
etc.
This form is called Roster or Braces form . In this form we make a list of the elements of the set
and put it within braces { }.
Instead of listing we could describe them as follows :
A = the set of vowels in the alphabet
B = The set of even numbers between 2 and 10 both inclusive.
C = The set of all possible arrangements of the letters p, q and r
D = The set of odd digits between 1 and 9 both inclusive.
E = The set of roots of the equation x2–3x + 2 = 0
Set B, D and E can also be described respectively as
B = {x : x = 2m and m being an integer lying in the interval 0 < m < 6}
D = {2x – 1 : 0 < x < 6 and x is an integer}
E = {x : x2 – 3x + 2 = 0}
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This form is called set-Builder or Algebraic form or Rule Method. This method of writing the
set is called Property method. The symbol : or/reads 'such that'. In this method , we list the
property or properties satisfied by the elements of the set.
We write, {x:x satisfies properties P } . This means, "the set of all those x such that x satisfies the
properties P"
A set may contain either a finite or an infinite number of members or elements. When the
number of members is very large or infinite it is obviously impractical or impossible to list them
all. In such case.
we may write as :
N = The set of natural numbers = {1, 2, 3…..}
W = The set of whole numbers = { 0, 1, 2, 3,…)
etc.
I. The members of a set are usually called elements, In A = {a,e,i,o,u}, a is an element and we
write aÎ A i.e. a belongs to A. But 3 is not an element of B = {2, 4, 6, 8, 10} and we write
3∉B. i.e. 3 does not belong to B.
II. If every element of a set P is also an element of set Q we say that P is a subset of Q. We write
P Ì Q . Q is said to be a superset of P. For example {a, b} Ì {a, b, c}, {2, 4, 6, 8, 10} Ì N. If
there exists even a single element in A , which is not in B then A is not a subset of B
III. If P is a subset of Q but P is not equal to Q then P is called a proper subset of Q .
IV. F has no proper subset.
IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnn::::: {3} is a proper subset of { 2, 3, 5}. But { 1, 2 } is not a subset of { 2, 3, 5} .
Thus if P = { 1, 2 } and Q = { 1, 2 ,3} then P is a subset of Q but P is not equal to Q . So , P is a
proper subset of Q.
To give completeness to the idea of a subset, we include the set itself and the empty set. The
empty set is one which contains no element. The empty set is also known as nnnnnuuuuullllllllll ooooorrrrr vvvvvoooooiiiiiddddd set
usually denoted by { } or Greek letter F , to be read as phi. For example the set of prime numbers
between 32 and 36 is a null set. The subsets of {1, 2, 3,} include {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1},
{2}, {3} and { }
A set containing n elements has 2n subsets. Thus a set containing 3 elements has
23 (=8) subsets. A set containing n elements has 2n –1 proper subsets. Thus a set containing 3
elements has 23 –1 ( =7) subsets. The proper subsets of { 1,2,3} include
{1, 2} ,{1, 3},{2, 3} ,{1},{2} ,{3] , { } .
Suppose we have two sets A and B. The intersection of these sets, written as AI B contains
those elements which are in A and are also in B.
For example A = {2, 3, 6, 10, 15}, B = {3, 6, 15, 18, 21, 24} and C = { 2, 5, 7},
we have A∩B = { 3, 6, 15}, A∩C = {2}, B∩C = Φ, where the intersection of B and C is empty
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set. So, we say B and C are disjoint sets since they have no common element. Otherwise sets
are called overlapping or intersecting sets. The union of two sets, A and B, written as AUB
contain all these elements which are in either A or B or both.
So AUB = {2, 3, 6, 10, 15, 18, 21, 24 }
AUC = {2, 3, 5, 6, 7, 10, 15}
A set which has at least one element is called non-empty set . Thus the set { 0 } is non-empty set.
It has one element say 0.
SSSSSiiiiinnnnngggggllllleeeeetttttooooonnnnn SSSSSeeeeettttt ::::: A set containing one element is called Singleton Set. For example
{ 1 } is a singleton set, whose only member is 1.
EEEEEqqqqquuuuuaaaaalllll SSSSSeeeeettttt ::::: Two sets A & B are said to be equal, written as A = B if every element of A is in B
and every element of B is in A.
IIIIIlllllllllluuuuussssstttttrrrrraaaaatttttiiiiiooooonnnnn::::: If A = { 2, 4, 6 } and B = { 2, 4, 6 } then A = B.
RRRRReeeeemmmmmaaaaarrrrrkkkkksssss ::::: (I) The elements of a set may be listed in any order.
Thus, { 1, 2, 3} = { 2, 1, 3 } = { 3, 2, 1 } etc.
(II) The repetition of elements in a set is meaningless.
EEEEExxxxxaaaaammmmmpppppllllleeeee ::::: {x : x is a letter in the word "follow" } = { f,o,l,w}
Φ
EEEEExxxxxaaaaammmmmpppppllllleeeee ::::: Show that , {0} and 0 are all different.
Φ
SSSSSooooollllluuuuutttttiiiiiooooonnnnn::::: Since is a set containing no element at all; {0} is a set containing one element,
namely 0. And 0 is a number , not a set.
Hence F ,{0} and 0 are all different.
The set which contains all the elements under consideration in a particular problem is called
the universal set denoted by S. Suppose that P is a subset of S. Then the complement of P,
written as Pc (or P') contains all the elements in S but not in P. This can also be written as S – P
or S ~ P. S – p = {x : x Î s, x Ï p}.
For example let S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
P = {0, 2, 4, 6, 8}
Q = {1, 2, 3, 4, 5)
Then P' = {1, 3 ,5 ,7, 9} and Q'= {0 , 6 , 7, 8, 9}
Also P∪Q = {0, 1, 2, 3, 4, 5, 6, 8}, (P∪Q)1 = {7, 9}
P∩Q = {2, 4}
P∪Q' = {0 , 2 , 4 ,6 ,7 , 8 ,9 }, (P∩Q)' = {0, 1, 3, 5, 6, 7, 8, 9}
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P'∪Q' = { 0,1, 3, 5, 6, 7, 8, 9}
P'∩Q' = {7, 9}
∩ ∩
So it can be noted that (PUQ)' = P' Q' and (P Q)' = P'UQ'. These are known as De Morgan’s
laws.
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We may represent the above operations on sets by means of Euler -Venn diagrams.
P
S
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0
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(a)
FFFFFiiiiiggggg..... 11111
Thus Fig. 1(a) shows a universal set S represented by a rectangular region and one of its subsets
P represented by a circular shaded region.
The un-shaded region inside the rectangle represents P'.
Figure 1(b) shows two sets P and Q represented by two intersecting circular regions. The total
shaded area represents PUQ, the cross - hatched "intersection" represents P∩Q.
The number of distinct elements contained in a finite set A is called its cccccaaaaarrrrrdddddiiiiinnnnnaaaaalllll nnnnnuuuuummmmmbbbbbeeeeerrrrr. It is
denoted by n( A ). For Example , the number of elements in the set R = {2, 3, 5, 7} is denoted by
n(R). This number is called the cardinal number of the set R.
A B
Thus n(AUB) = n(A) + n(B) – n(A∩ B)
∩
If A and B are disjoint sets, then n(AUB) = n(A) + n(B) as n (A B) = 0
A B
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P Q
R
For three sets P, Q and R
n(PUQUR) = n(P) + n(Q)+n(R) – n(P∩Q) – n(Q∩R) – n(P∩R) + n(P∩Q∩R)
When P,Q and R are disjoint sets
∪ ∪
n(P Q R)
= n(P) + n(Q) + n(R)
Illustration : If A = { 2, 3, 5, 7 } , then n (A) = 4
Equivalent Set : Two finite sets A & B are said to be equivalent if n (A) = n(B).
Clearly, equal sets are equivalent but equivalent sets need not be equal.
Illustration : The sets A = { 1, 3, 5 } and B = { 2, 4, 6} are equivalent but not equal.
Here n (A) = 3 = n (B) so they are equivalent sets. But the elements of A are not in B. Hence they
are not equal though they are equivalent.
Power Set : The collection of all possible subsets of a given set A is called the power set of A , to
be denoted by P(A).
Illustration : (I) If A = {1, 2, 3} then
P(A) = { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3} , Φ}
(II) If A = { 1, {2}} , we may write A = { 1, B} when B = {2} then
P(A) = {Φ, {1} ,{B } , { 1,B}} = {Φ, { 1} ,{ {2} } , { 1,{2 }}}
Exercise 7 (A)
Choose the most appropriate option or options (a), (b) (c) and (d)
1. The number of subsets of the set {2, 3, 5} is
(a) 3 , (b) 8, (c) 6, (d) none of these,
2. The number of subsets of a set containing n elements is
(a) 2n (b) 2–n (c) n (d) none of these
3. The null set is represented by
(a){Φ } (b) { 0 } (c) Φ (d) none of these
4. A = {2, 3, 5, 7} , B { 4, 6, 8, 10} then A∩B can be written as
(a) { } (b) {Φ} (c) (AUB)' (d) None of these
5 The set {x|0<x<5} represents the set when x may take integral values only
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(a) {0, 1, 2, 3, 4, 5 } (b) {1, 2, 3, 4 } c) {1, 2, 3, 4, 5 } (d) none of these
6. The set {0, 2, 4, 6, 8, 10} can be written as
(a) {2x | 0<x <5} (b) {x : 0<x<5} (c) {2x : 0<x<5} (d) none of these
If P = {1, 2, 3, 5, 7}, Q = {1, 3, 6, 10, 15}, Universal Set S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15}
7. The cardinal number of PI Q is
(a) 3, (b) 2 (c) 0 (d) none of these
8. The cardinal number of PUQ is
(a) 10, (b) 9, (c) 8, (d) none of these
9. n (P1) is
(a) 10, (b) 5, (c) 6, (d) none of these
10. n(Q1) is
(a) 4, (b) 10, (c) 4, (d) none of these
11. The set of cubes of the natural number is
(a) a finite set, (b) an infinite set, (c) a null set (d) none of these
12. The set {2x|x is any positive rational number } is
(a) an infinite set, (b) a null set, (c) a finite set, (d) none of these
13. {1– (–1)x} for all integral x is the set
(a) {0}, (b) {2}, (c) {0,2} (d) none of these
14. E is a set of positive even number and O is a set of positive odd numbers, then E∪O is a
(a) set of whole numbers, (b) N, (c) a set of rational number, (d) none of these
15. If R is the set of positive rational number and E is the set of real numbers then
(a) R C E, (b) R C E (c) E C R (d) none of these
16. If N is the set of natural numbers and I is the set of positive integers, then
(a) N = I, (b) N Ì I, (c) N CI, (d) none of these
17.cIf I is the set of isosceles triangles and E is the set of equilateral triangles, then
(a) IÌ E, (b) EÌ I, (c) E=I (d) none of these
18. If R is the set of isosceles right angled triangles and I is set of isosceles triangles, then
(a) R = I (b) R ÌI, (c) RÌ I (d) none of these
19. {n(n+1)/2 : n is a positive integer} is
(a) a finite set (b) an infinite set (c) is an empty set (d) none of these
20. If A = {1, 2, 3, 5, 7}, and B = {x2 : x ∈ A}
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SETS, FUNCTIONS AND RELATIONS
(a) n(b) = n(A), (b) n(B) > n(A) (c) n(A)= n(B) (D) n(A)<n(B)
21. A∪A is equal to
a) A, (b) E, (c) φ (d) none of these
22. A∩A is equal to
(a) φ (b) A, (c) E, (d) none of these
23. (A∪B)' is equal to
(a) (A∩B)' (b) A∪B' (c) A'∩B', (d) none of these
24. (A∩B)' is equal to
(a) (A'∪B)' (b) A'∪ B' (c) A'∩B', (d) none of these
25. A∪E is equal to (E is a superset of A)
(a) A, (b) E, (c) φ, (d) none of these
26. A∩E is equal to
(a) A (b) E, (c) φ (d) none of these
27. E∪E is equal to
(a) E, (b) φ, (c) 2E, (d) none of these
28. A∩E' is equal to
(a) E (b) φ , (c) A, (d) none of these
29. A∩F is equal to
(a) A (b) E (c) φ (d) none of these
30. AUA' is equal to
(a) E (b) φ, (c) A, (d) none of these
31. If E = {1, 2, 3, 4, 5, 6, 7, 8, 9}, the subset of E satisfying 5 + x > 10 is
(a) {5, 6, 7, 8, 9} (b) {6, 7, 8, 9}, (c) {7, 8, 9}, (d) none of these
32. If A∆ B = (A–B)∪(B–A) and A = {1, 2, 3, 4}, B = {3,5,7} than A∆B is
(a) {1, 2, 4, 5, 7} (b) {3} (c) {1, 2, 3, 4, 5, 7} (d) none of these
[Hint : If A and B are any two sets, then
A - B = { x : x ∈A, x ∉ B}.
i.e. A - B Contains all elements of A but not in B] .
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7.3 PRODUCT SETS
Ordered Pair : Two elements a and b, listed in a specific order, form an ordered pair, denoted
by (a, b).
Cartesian Product of sets : If A and B are two non-empty sets, then the set of all ordered pairs
(a, b) such that a belongs to A and b belongs to B , is called the Cartesian product of A and B,
to be denoted by A × B.
∈ ∈
Thus, A × B = { ( a, b) : a A and b B }
Φ Φ
If A = or B = F , we define A × B =
Illustration : Let A = {1, 2, 3}, B = {4, 5}
Then A × B = { (1, 4), (1, 5), (2, 4) (2, 5), (3, 4), (3, 5) }
Example: If A × B = { ( 3, 2 ) , (3, 4) , (5, 2), (5, 4) } , find A and B.
Solution: Clearly A is the set of all first co-ordinates of A × B , while B is the set of all second
co-ordinates of elements of A × B .
Therefore A = { 3, 5 } and B = {2 , 4 }
Example : Let P = {1, 3, 6} and Q { 3, 5}
The product set P × Q = {(1, 3), (1, 5), (3, 3), (3, 5), (6, 3), (6, 5)} .
Notice that n(P×Q) = n(P) × n(Q) and ordered pairs (3,5) and (5,3) are not equal.
and Q×P = {(3, 1), (3, 3), (3, 6), (5, 1), (5, 3), (5, 6)}
≠
So P×Q Q×P; but n(P×Q) = n(Q×P).
Illustration: Here n(P) = 3 and n(Q ) = 2 , n(P × Q) = 6 Hence n(P×Q) = n(p) × n(Q). and
n(P × Q ) = n(Q × P) = 6.
Y
(1,5) (3,5)
5 (6,5)
We can represent the product set of 4
ordered pairs by points in the XY plane. 3
(6,3)
2 (1,3) (3,3)
1
X
1 2 3 4 5 6
If X=Y= the set of all natural numbers, the product set X, Y is represented by an infinite equal
lattice of points in the first quadrant of the XY plane.
MMMMMAAAAATTTTTHHHHHSSSSS 77777.....99999
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SETS, FUNCTIONS AND RELATIONS
7.4 RELATIONS AND FUNCTIONS
Any subset of the product set XY is said to define a relation from X to Y and any relation from
X to Y in which no two different ordered pairs have the same first element is called a function.
Let A and B be two non-empty sets. Then, a rule or a correspondence f which associates to
each element x of A, a unique element, denoted by f(x) of B , is called a function or mapping
from A to B and we write f : A→B
The element f(x) of B is called the image of x, while x is called the pre-image of f (x).
7.5 DOMAIN & RANGE OF A FUNCTION
Let f : A→B, then A is called the domain of f, while B is called the co-domain of f.
∈
The set f(A) = { f (x) : x A } is called the range of f.
Illustration : Let A = { 1, 2, 3, 4 } and B ={1, 4, 9, 16, 25 }
We consider the rule f(x) = x2 . Then f(1) = 1 ; f(2) =4 ; f (3) = 9 & f (4) = 16.
Then clearly each element in A has a unique image in B.
So, f : A → B : f (x) = x2 is a function from A to B.
Here domain ( f) = { 1, 2, 3, 4 } and range (f) = {1, 4, 9, 16 }
Example : Let N be the set of all natural numbers. Then , the rule
N N
f
3 6
2 4
1 2
f : N → N : f(x) = 2x , for all x ∈ N
is a function from N to N , since twice a natural number is unique.
Now, f (1) = 2; f (2) = 4; f(3) = 6 and so on.
Here domain (f) = N = {1, 2 ,3, 4 ,………..} ; range ( f) = { 2, 4, 6,………..}
This may be represented by the mapping diagram or arrow graph .
7.6 VARIOUS TYPES OF FUNCTION
One - one Function : Let f : A→B. If different elements in A have different images in B, then f
is said to be a one-one or an injective function or mapping.
Illustration : (i) Let A = { 1, 2, 3} and B = { 2, 4, 6}
Let us consider f : A→B : f(x) = 2x.
Then f(1) = 2; f(2) =4 ; f (3) = 6.
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Clearly, f is a function from A to B such that different elements in A have different images in B.
Hence f is one -one.
∈
Remark : Let f : A→B and let x , x A.
1 2
Then x = x implies f(x ) = f(x ) is always true.
1 2 1 2
But f(x ) = f(x ) implies x = x is true only when f is one-one.
1 2 1 2
(ii) let x={1, 2, 3, 4} and y={1, 2, 3}, then the subset {(1, 2), (1, 3), (2, 3)} defines a relation on x.y.
X Y
1 1
2
2
3
Notice that this particular subset contains all the ordered pair in x.y for which the X element
(x) is less than the Y element (y). So in this subset we have X<Y and the relation between the
set, is "less than". This relation is not a function as it includes two different ordered pairs (1,2),
(1,3) have same first element.
X.Y={(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)
(3, 1), (3, 2), (3, 3),(4, 1), (4, 2), (4, 3)}
The subset {(1, 1), (2, 2), (3, 3)} defines the function y= x as different ordered pairs of this subset
have different first element.
Onto or Surjective Functions : Let f : A→B. If every element in B has at least one pre-image in
A , then f is said to be an onto function.
∈
If f is onto , then corresponding to each y B, we must be able to find at least one element
x Î A such that y = f (x)
Clearly, f is onto if and only if range (f) = B
Illustration : Let N be the set of all natural numbers and E be the set of all even natural
numbers. Then, the function
f : N → E : f (x) = 2x , for all x ∈ N
∈
is onto, since each element of E is of the form 2x , where x N and the same is the
∈
f-image of x N.
1 1
2 2
3 3
X Y
Represented on a mapping diagram it is a one-one mapping of X onto Y.
Bijection Function : A one-one and onto function is said to be bijective.
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SETS, FUNCTIONS AND RELATIONS
A bijective function is also known as a one-to-one correspondence.
Identity Function : Let A be a non-empty set . Then, the function I defined by
I : A → A : I (x) = x for all x ∈ A is called an identity function on A.
It is a one-to-one onto function with domain A and range A.
Into Functions: Let f : A → B. There exists even a single element in B having no pre-image in A
, then f is said to be an into function.
Illustration : Let A = { 2, 3, 5, 7 } and B = { 0, 1, 3, 5, 7} . Let us consider f : A → B;
f (x) = x – 2. Then f(2) = 0; f(3) = 1 ; f (5) =3 & f(7) = 5.
It is clear that f is a function from A to B .
Here there exists an element 7 in B, having no pre-mage in A.
So, f is an into function.
Constant Function: Let f : A → B, defined in such a way that all the elements in A have the
same image in B, then f is said to be a constant function.
Illustration: Let A = { 1, 2, 3 } and B = { 5, 7, 9} . Let f : A → B : f (x) = 5 for all x ∈ A.
Then, all the elements in A have the same image namely 5 in B.
So, f is a constant function.
Remark: The range set of a constant function is a singleton set.
Example: Another subset of x.y is {(1,3), (2,3), (3,3) , (4,3)}
X Y
1 1
2
4 2
3 3
This relation is a function (a constant function). It is represented on a mapping diagram and is
a many-one mapping of X into Y.
Let us take another subset {(4,1), (4,2), (4,3)} of X.Y. This is a relation but not a function. Here
different ordered pairs have same first element so it is not a function.
1
1
2
X > 2 Y
3
3
4
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Copyright -The Institute of Chartered Accountants of India
x y
1
5
2
3 4
Many–one mapping
This is an example of many - one mapping.
Equal Functions: Two functions f and g are said to be equal, written as f = g if they have the
same domain and they satisfy the condition f(x) = g(x), for all x.
A function may simply pair people and the corresponding seat numbers in a theatre. It may
simply associate weights of parcels with portal delivery charge or it may be the operation of
squaring , adding to doubling, taking the log of etc.
x x
f
A 101
B 236
C 300
The function f here assigning a locker number to each of the persons A, B and C. Names are
associated with or mapped on to, locker numbers under the function f.
We can write f : X → Y OR, f(x) = y OR, f(B) = 236
x y z
n g
1 0 3
2 1 4
4 2 5
8 3 6
This diagram shows the effect of two functions n and g on the sets X, Y and Z
n : X→Y and g : Y→Z
If x, y, z are corresponding elements of X ,Y and Z, we write n(x) = y and g(y) = z
Thus n(1) = 0 and g (0) = 3, so that g (n(1)) = g(0) = 3 we can write it as
g n(1) or g o n (1) = 3 But g(1) = 4 and n(g(1) )= n(4) = 2
So gn≠ ng (or, g o n ≠ n o g)
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Copyright -The Institute of Chartered Accountants of India
SETS, FUNCTIONS AND RELATIONS
The function gn or ng is called a composite function. As n(8) = 3, we say that 3 is the image of
8 under the mapping (or function) n.
Inverse Function: Let f be a one-one onto function from A to B. Let y be an arbitrary element
of B. Then f being onto, there exists an element x in A such that f (x) = y .
As f is one-one this x is unique.
Thus for each yÎB, there exists a unique element x ∈ A such that f (x) = y.
So, we may define a function, denoted by f –1 as:
f–1 : B→A : f–1 (y ) = x if and only if f (x) = y.
The above function f–1 is called the inverse of f.
A function is invertible if and only if f is one-one onto.
Remarks : If f is one -one onto then f–1 is also one-one onto.
Illustration : If f : A→B then f –1 : B→A.
Exercise 7(B)
Choose the most appropriate option/options (a), (b), (c) or (d)
1. If A = {x, y, z}, B = {p, q, r, s} Which of the relation on A.B are function.
(a) {n, p), (x, q), (y, r), (z, s)}, (b) {( x, s), (y, s), (z, s)}
(c) {(y, p), (y, q), (y, r),(z, s), (d) {(x, p), (y, r), (z, s)}
2. {(x,y)|x+y = 5} is a
(a) not a function (b) a composite function (c) one-one mapping (d) none of these
3. {( x , y)|x = 4} is a
(a) not a function (b) function (c) one-one mapping (d) none of these
4. {(x , y), y=x2} is
(a) not a function (b) a function (c) inverse mapping (d) none of these
5. {(x, y)|x<y} is
(a) not a function (b) a function (c) one-one mapping (d) none of these
6. The domain of {(1,7), (2,6)} is
(a) (1, 6) (b) (7, 6) (c) (1, 2) (d) {6, 7}
7. The range of {(3,0), (2,0), (1,0), (0,0)} is
(a) {0, 0} (b) {0} (c) {0, 0, 0, 0} (d) none of these
8. The domain and range of {(x,y) : Y = x2 } is
(a) (reals, natural numbers) (b) (reals, positive reals)
(c) (reals, reals) (d) none of these
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Copyright -The Institute of Chartered Accountants of India
9. Let the domain of x be the set {1}. Which of the following functions are equal to 1
(a) f(x) = x2, g(x) = x (b) f(a) = x, g(x) = 1–x
(c) f(x) = x2 +x +2 , g(x) = (x+ 1)2 (d) none of these
10. If f(x) = 1/1–x, f(–1) is
(a) 0 (b) ½ (c) 0 (d) none of these
11. If g(x) = (x–1)/x, g(–½) is
(a) 1 (b) 2 (c) 3/2 (d) 3
12. If f(x) = 1/1–x and g(x) = (x–1)/x, than fog(x) is
(a) x (b) 1/x (c) –x (d) none of these
13. If f(x) = 1/1–x and g(x) = (x–1)/x, then g of(x) is
(a) x–1 (b) x (c) 1/x (d) none of these
14. The function f(x) = 2x is
(a) one-one mapping (b) one-many
(c) many-one (d) none of these
15. The range of the function f(x) = log (1 + x) for the domain of real values of x when 0£ x
10
£ 9 is
(a) {0, –1} (b) {0, 1, 2} (c) {0.1} (d) none of these
16. The Inverse function f–1 of f(x) = 2x is
x
(a) 1/2x (b) (c) 1/x (d) none of these
2
17. If f(x) = x+3, g(x) = x2, then fog(x) is
(a) x2 + 3 (b) x2 + x +3 (c) (x+3)2 (d) none of these
18. If f(x) = x+3, g(x) = x2 then f(x).g(x) is
(a) (x+3)2 (b) x2+3 (c) x3+3x2 (d) none of these
19. The Inverse h–1 when h(x) = log x is
10
(a) log x (b) 10x (c) log (1/x) (d) none of these
10 10
20. For the function h(x) = 101+x the domain of real values of x where 0 < x < 9, the range is
(a) 10 < h(x) < 1010 (b) 0 < h(x) < 1010 (c) 0 < h(x) < 10
(d) none of these
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SETS, FUNCTIONS AND RELATIONS
Different types of relations
Let S = {a, b, c, ….} be any set then the relation R is a subset of the product set S×S
i) If R contains all ordered pairs of the form (a, a) in S×S, then R is called reflexive. In a
reflexive relation 'a' is related to itself .
For example, 'Is equal to' is a reflexive relation for a = a is true.
ii) If (a, b)∈R⇒(b,a)∈ R for every a, b∈S then R is called symmetric
For Example a = b⇒b = a. Hence the relation 'is equal to' is a symmetric relation.
iii) If (a, b)∈R and (b, c)∈R⇒(a, c)⇒R for every a, b, c, ∈ S then R is called transistive.
For Example a =b, b=c⇒a=c. Hence the relation 'is equal to' is a transitive relation.
A relation which is reflexive, symmetric and transitive is called an equivalence relation or simply
an equivalence. 'is equal to' is an equivalence relation.
Similarly, the relation " is parallel to " on the set S of all straight lines in a plane is an equivalence
relation.
Illustration : The relation " is parallel to " on the set S is
∈
(1) reflexive, since a || a for a S
(2) symmetric, since a || b ⇒ b || a
(3) transitive, since a || b , b || c ⇒ a || c
Hence it is an equivalence relation.
Domain & Range of a relation : If R is a relation from A to B, then the set of all first co-
ordinates of elements of R is called the domain of R, while the set of all second co-ordinates of
elements of R is called the range of R.
So, Dom (R) = { a : (a, b) ∈ R } & Range ( R) = { b : (a, b) ∈ R}
Illustration: Let A = { 1, 2, 3} and B = { 2, 4, 6}
Then A × B = {(1,2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)}
By definition every subset of A × B is a relation from A to B.
Thus, if we consider the relation
R = { (1, 2), (1, 4), (3, 2), (3, 4)} then Dom ( R ) = { 1,3} and Range ( R )= { 2, 4}
From the product set X. Y = {(1, 3), (2, 3), (3, 3), (4, 3), (2, 2), (3, 2), (4, 2), (1, 1), (2, 1), (3, 1),
(4, 1)}, the subset {(1, 1), (2, 2), (3, 3)} defines the relation 'Is equal to' , the subset {(1, 3), (2, 3),
(1, 2)} defines 'Is less than' , the subset {(4, 3), (3, 2), (4, 2), (2, 1), (3, 1), (4, 1)} defines 'Is greater
than' and the subset {(4, 3), (3, 2), (4, 2), (2, 1), (3, 1), (4, 1), (1, 1), (2, 2) (3, 3)} defines to greater
'In greater than or equal'.
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Copyright -The Institute of Chartered Accountants of India
Illustration : Let A = {1, 2, 3} and b = {2, 4, 6}
Then A × B = {(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)}
If we consider the relation = {(1, 2), (1, 4), (3, 4)} then Dom (R) = {1, 3} and
Range = {2, 4} Here the relation "Is less than".
∈
Identity Relation: The relation I = {(a, a) : a A} is called the identity relation on A.
Illustration: Let A = {1, 2, 3} then I = {(1, 1), (2, 2), (3, 3)}
Inverse Relation: If R be a relation on A , then the relation R–1 on A , defined by
∈
R–1 = { ( b, a) : (a, b) R} is called an inverse relation on A.
Clearly , Dom (R–1 ) = Range (R) & Range (R–1) = Dom ( R ).
Illustration: Let A = {1, 2, 3} and R = {(1, 2), (2, 2), (3, 1), (3, 2)}
Then R being a subset of a × a , it is a relation on A. Dom (R) = {1, 2, 3} and Range (R) = {2,1}
Now, R–1 = {(2, 1), (2, 2), (1, 3), (2, 3)} Here, Dom (R–1) = {2, 1} = Range (R) and
Range (R–1 ) = {1, 2, 3} = Dom (R).
Illustration: Let A = {1, 2, 3}, then
(i) R1 = {(1, 1), (2, 2), (3, 3), (1, 2)}
∈
Is reflexive and transitive but not symmetric, since (1, 2) R but (2, 1) does not belongs to R .
1 1
(ii) R2 = {(1, 1), (2, 2), (1, 2), (2, 1)}
is symmetric and transitive but not reflexive, since (3, 3) does not belong to R .
2
(iii) R3 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}
is reflexive and symmetric but not transitive , since (1, 2) ∈ R3 & (2, 3) ∈ R3 but
(1, 3) does not belong to R3.
Problems and solution using Venn Diagram
1. Out of a group of 20 teachers in a school, 10 teach Mathematics, 9 teach Physics and 7
teach Chemistry. 4 teach Mathematics and Physics but none teach both Mathematics and
Chemistry. How many teach Chemistry and Physics? How many teach only Physics ?
P
C
X
9-4-X
7-X
4
10-4=6
M
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Copyright -The Institute of Chartered Accountants of India
SETS, FUNCTIONS AND RELATIONS
Let x be the no. of teachers who teach both Physics & Chemistry.
9–4–x+6+7–x+4+x=20
or 22–x=20
or x=2
Hence, 2 teachers teach both Physics and Chemistry and 9–4–2 = 3 teachers teach only Physics.
2. A survey shows that 74% of the Indians like grapes, whereas 68% like bananas.
What percentage of the Indians like both grapes and bananas?
Solution: Let P & Q denote the sets of Indians who like grapes and bananas respectively. Then
n(P ) = 74, n(Q) = 68 and n(P∩Q) = 100.
We know that n( P∩Q) = n(P) + n(Q) – n( P∩Q ) = 74 + 68 – 100 = 42.
Hence, 42% of the Indians like both grapes and bananas.
3. In a class of 60 students, 40 students like Maths, 36 like Science, and 24 like both the
subjects. Find the number of students who like
(i) Maths only. (ii) Science only (iii) either Maths or Science
(iv) neither Maths nor Science.
Solution: Let M = students who like Maths and S = students who like Science
Then n( M) = 40, n(S) = 36 and n (M∩S ) = 24
Hence, (i) n(M) – n(M∩S) = 40 – 24 = 16 = number of students like Maths only.
(ii) n( S ) – n(M∩S) = 36 – 24 = 12 = number of students like Science only.
(iii) n(M∩S) = n(M) + n(S) – n(M∩S) = 40 + 36 – 24 = 52 = number of students who like either
Maths or Science.
( iv) n(M∩S)c = 60 – n(M∩S ) = 60 – 52 = 8 = number of students who like neither Maths nor
Science.
Exercise 7C
Choose the most appropriate option/options (a), (b), (c) or (d)
1. "Is smaller than" over the set of eggs in a box is
a) Transitive (T) (b) Symmetric (S) (c) Reflexive (R) (d) Equivalence (E)
2. "Is equal to" over the set of all rational numbers is
(a) (T) (b) (S) (c) (R) (d) E
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Copyright -The Institute of Chartered Accountants of India
3. "has the same father as" …… over the set of children
(a) R (b) S (c) T (d) none of these
4. "is perpendicular to " over the set of straight lines in a given plane is
(a) R (b) S (c) T (d) E
5. "is the reciprocal of" …….. over the set of non-zero real numbers is
(a) S (b) R (c) T (d) none of these
6. {(x,y)/x∈×,y∈y,y =x} is
(a) R (b) S (c) T (d) none of these
7. {(x,y) / x + y = 2x where x and y are positive integers}, is
(a) R (b) S (c) T (d) E
8. "Is the square of" over n set of real numbers is
(a) R (b) S (c) T (d) none of these
9. If A has 32 elements, B has 42 elements and A∩B has 62 elements, the number of elements
in A∩B is
(a) 12 (b) 74 (c) 10 (d) none of these
10 In a group of 20 children, 8 drink tea but not coffee and 13 like tea. The number of children
drinking coffee but not tea is
(a) 6 (b) 7 (c) 1 (d) none of these
11 The number of subsets of the sets {6, 8, 11} is
(a) 9 (b) 6 (c) 8 (d) none of these
12. The sets V = {x / x+2=0}, R={x / x2+2x=0} and S = {x : x2+x–2=0} are equal to one another
if x is equal to
(a) –2 (b) 2 (c) ½ (d) none of these
13. If the universal set E = {x |x is a positive integer <25} , A = {2, 6, 8, 14, 22}, B = {4, 8, 10, 14}
then
ϕ
(a) (A∩B)'=A'∩ B' (b) (A∩B)'= A'∩B' (c) (A'∩B)'= (d) none of these
14. If the set P has 3 elements, Q four and R two then the set P×Q×R contains
(a) 9 elements (b) 20 elements (c) 24 elements (d) none of these
15. Given A = {2, 3}, B = {4, 5}, C = {5, 6} then A × (B∩C) is
(a) {(2, 5), (3, 5)} (b) {(5, 2), (5, 3)} (c) {(2, 3), (5, 5)} (d) none of these
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Copyright -The Institute of Chartered Accountants of India
SETS, FUNCTIONS AND RELATIONS
16. A town has a total population of 50,000. Out of it 28,000 read the newspaper X and 23000
read Y while 4000 read both the papers. The number of persons not reading X and Y both
is
(a) 2000 (b) 3000 (c) 2500 (d) none of these
17. If A = { 1, 2, 3, 5, 7} and B = {1, 3, 6, 10, 15}. Cardinal number of A~B is
(a) 3 (b) 4 (c) 6 (d) none of these
18. Which of the diagram is graph of a function
Y Y Y Y
(a) (b) (c) (d)
x x x x
19. At a certain conference of 100 people there are 29 Indian women and 23 Indian men. Out
of these Indian people 4 are doctors and 24 are either men or doctors. There are no foreign
doctors. The number of women doctors attending the conference is
(a) 2 (b) 4 (c) 1 (d) none of these
20. Let A = {a, b}. Set of subsets of A is called power set of A denoted by P(A). Now n(P(A) is
(a) 2 (b) 4 (c) 3 (d) none of these
21. Out of 2000 employees in an office 48% preferred Coffee (c), 54% liked (T), 64% used to
smoke (S). Out of the total 28% used C and T, 32% used T and S and 30% preferred C and
S, only 6% did none of these. The number having all the three is
(a) 360 (b) 300 (c) 380 (d) none of these
22. Referred to the data of Q. 21 the number of employees having T and S but not C is
(a) 200 (b) 280 (c) 300 (d) none of these
23. Referred to the data of Q. 21. the number of employees preferring only coffee is
(a) 100 (b) 260 (c) 160 (d) none of these
24. If f(x) = x+3, g(x) = x2, then gof(x) is
(a) (x+3)2 (b) x2+3 (c) x2(x+3), (d) none of these
25. If f(x) = 1/1–x, then f-1(x) is
(a) 1–x (b) x–1/x (c) x/x–1 (d) none of these
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ANSWERS
Exercise 7(A)
1. b 2. a 3. c 4. a 5. b 6. c 7. b 8. c
9. a 10. b 11. b 12. a 13. c 14. b 15 b 16. a
17. b 18. c 19. b 20. a 21. a 22. b 23. c 24. b
25. b 26. a 27. a 28. b. 29. c 30. b 31. b. 32. a
Exercise 7(B)
1. b,d 2. c 3. a 4. b 5. a 6. c 7. b 8. b
9. a 10. b 11. d 12. a 13. b 14. a 15. c 16. b
17. a 18. c 19. b 20. a.
Exercise 7(C)
1. T 2. a,b,c,d 3. a,b,c 4. b 5. a 6. a,b,c 7. a,b 8. d
9. a 10. b 11. c 12. a 13. a 14. c 15. a 16. b
17. a 18. b 19. c 20. b 21. a 22. b 23. c 24. a
25. b
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SETS, FUNCTIONS AND RELATIONS
ADDITIONAL QUESTION BANK
⊂ ∉ ⊃ ⊄
1. Following set notations represent: – A B; x A; A B; {0}; A B
(A) A is a proper subset of B; x is not an element of A; A contains B; singleton with an
only element zero; A is not contained in B
(B) A is a proper subset of B; x is an element of A; A contains B; singleton with an only
element zero; A is contained in B
(C) A is a proper subset of B; x is not an element of A; A does not contains B; contains
elements other than zero; A is not contained in B
(D) None
2. Represent the following sets in set notation: – Set of all alphabets in English language set
of all odd integers less than 25 set of all odd integers set of positive integers x satisfying the
equationx2+5x+7=0: -
(A) A={x:x is an alphabet in English}, I={x:x is an odd integer>25}, I={2, 4, 6, 8 ….}
I={x: x2+5x+7=0}
(B) A={x:x is an alphabet in English}, I={x:x is an odd integer<25}, I={1, 3, 5, 7 ….}
I={x:x2+5x+7=0}
(C) A={x:x is an alphabet in English}, I={x:x is an odd integer £ 25}, I={1, 3, 5, 7 ….}
I={x:x2+5x+7=0}
(D) None
3. Re-write the following sets in a set builder form: - A={a, e, i, o, u} B={1, 2, 3, 4 ….} C is a set
of integers between –15 and 15.
(A) A={x:x is a consonant} B={x:x is an irrational number} C={x: –15<x<15∧ x is a fraction}
(B) A={x:x is a vowel} B={x:x is a natural number} C={x: –15³x³15∧ x is a whole number}
(C) A={x:x is a vowel} B={x:x is a natural number} C={x: –15 <x<15∧ x is a whole number}
(D) None
4. If V={0, 1, 2, …9}, X={0, 2, 4, 6, 8}, Y={3, 5, 7} and Z={3 7} then
Y∪Z, (V∪Y) ∩X, (X∪Z)∪V are respectively: –
(A) {3, 5, 7}, {0, 2, 4, 6, 8}, {0, 1, 2, …9} (B) {2, 4, 6}, {0, 2, 4, 6, 8}, {0, 1, 2, …9}
(C) {2, 4, 6}, {0, 1, 2, …9}, {0, 2, 4, 6, 8} (D) None
5. In question No.(4) (X∪Y) ∩Z and (φ UV) I φ are respectively: –
(A) {0, 2, 4, 6, 8}, φ (B) {3, 7}, φ (C) {3, 5, 7}, φ (D) None
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6. If V={x: } R={x: } and S={x: } then V, R, S are equal for the value of x equal to ______.
(A) 0 (B) –1 (C) –2 (D) None
7. What is the relationship between the following sets? A={x:x is a letter in the word flower}
B={x:x is a letter in the word flow} C={x:x is a letter in the word wolf} D={x:x is a letter in
the word follow}
(A) B=C=D and all these are subsets of the set A
(B) B=C≠D (C) B≠C≠D (D) None
8. Comment on the correctness or otherwise of the following statements: – (i) {a, b, c}={c, b,
a} (ii) {a, c, a, d, c, d} ⊂ {a, c, d} (iii) {b} ∈ {{b}} (iv) {b} ⊂ {{b}} and φ ⊂ {{b}}.
(A) Only (iv) is incorrect (B) (i) (ii) are incorrect
(C) (ii) (iii) are incorrect (D) All are incorrect
9. If A={a, b, c}, B={a, b}, C = {a, b, d}, D={c, d} and E={d} state which of the following
statements are correct: – (i) B ⊂ A (ii) D ≠ C (iii) C ⊃ E (iv) D E (v) D ⊂ B (vi) D = A (vii) B
⊄ C (viii) E ⊂ A (ix) E ⊄ B (x) a ∈ A (xi) a ⊂ A (xii) {a} ∈ A (xiii) {a} ⊂ A
(A) (i) (ii) (iii) (ix) (x) (xiii) only are correct (B) (ii) (iii) (iv) (x) (xii) (xiii) only are correct
(C) (i) (ii) (iv) (ix) (xi) (xiii) only are correct (D) None
10. Let A = {0}, B = {0 1}, C = φ, D = {φ}, E = {x|x is a human being 300 years old}, F = {x|x ∈
A and x ∈ B} state which of the following statements are true: – (i) A ⊂ B (ii) B = F (iii) C
⊂ D (iv) C = E (v) A = F (vi) F = 1 and (vii) E = C = D
(A) (i) (iii) (iv) and (v) only are true (B) (i) (ii) (iii) and (iv) are true
(C) (i) (ii) (iii) and (vi) only are true (D) None
11. If A = {0, 1} state which of the following statements are true: – (i) {1} ⊂ A (ii) {1} ∈ A (iii) φ
∈ A (iv) 0 ∈ A (v) 1 ⊂ A (vi) {0} ∈ A (vii) φ ⊂ A
(A) (i) (iv) and (vii) only are true (B) (i) (iv) and (vi) only are true
(C) (ii) (iii) and (vi) only are true (D) None
12. State whether the following sets are finite infinite or empty: – (i) X = {1, 2, 3, …..500} (ii) Y
= {y:y =a2; a is an integer } (iii) A = {x:x is a positive integer multiple of 2} (iv) B = {x:x is
an integer which is a perfect root of 26<x<35}
(A) finite infinite infinite empty (B) infinite infinite finite empty
(C) infinite finite infinite empty (D) None
13. If A = {1, 2, 3, 4} B = {2, 3, 7, 9} and C = {1, 4, 7, 9} then
(A) A ∩ B ≠ φ B ∩ C ≠ φ A ∩ C ≠ φ but A ∩ B ∩ C = φ
(B) A ∩ B = φ B ∩ C = φ A ∩ C = φ A ∩ B ∩ C = φ
(C) A ∩ B ≠ φ B ∩ C ≠ φ A ∩ C ≠ φ A ∩ B ∩ C ≠ φ
(D) None
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SETS, FUNCTIONS AND RELATIONS
14. If the universal set is X = {x:x ∈ N 1≤ x ≤ 12} and A = {1, 9, 10} B = {3, 4, 6, 11, 12} and C =
{2, 5, 6} are subsets of X the set A ∪ (B ∩ C) is _______.
(A) {3, 4, 6, 12} (B) {1, 6, 9, 10} (C) {2, 5, 6, 11} (D) None
15. As per question No.(14) the set (A ∪ B) ∩ (A ∪ C) is _________.
(A) {3, 4, 6, 12} (B) {1, 6, 9, 10} (C) {2, 5, 6, 11} (D) None
16. A sample of income group of 1172 families was surveyed and noticed that for income
groups < Rs.6000/-, Rs.6000/- to Rs.10999/-, Rs.11000/-, to Rs.15999/-, Rs.16000 and
above No. TV set is available to 70, 50, 20, 50 families one set is available to 152, 308, 114,
46 families and two or more sets are available to 10, 174, 84, 94 families.
If A = {x|x is a family owning two or more sets}, B = {x|x is a family with one set,
}C = {x|x is a family with income less than Rs.6000/-}, D = {x|x| is a family with income
Rs.6000/- to Rs.10999/-}, E = {x|x is a family with income Rs. 11000/- to Rs. 15999/-},
find the number of families in each of the following sets (i) C ∩ B
(ii) A ∪ E
(A) 152, 580 (B) 152, 20 (C) 152, 50 (D) 152, 496
17. As per question No.(16) find the number of families in each of the following sets: –
(i) (A ∪ B)′ ∩ E (ii) (C ∪ D ∪ E) ∩ (A ∪ B)′
(A) 20, 50 (B) 152, 20 (C) 152, 50 (D) 20, 140
18. As per question No.(16) express the following sets in set notation: –
(i) {x|x is a family with one set and income of less than Rs.11000/-}
(ii) {x|x is a family with no set and income over Rs.16000/-}
(A) (C ∪ D) ∩ B (B) (A ∪ B)′ ∩ (C′ ∪ D′ ∪ E′)
(C) Both (D) None
19. As per question No.(16) express the following sets in set notation: –
(i) {x|x is a family with two or more sets or income of Rs.11000/- to Rs.15999/-}
(ii) {x|x is a family with no set}
(A) (A ∪ E) (B) (A ∪ B)′ (C) Both (D) None
20. If A = {a, b, c, d} list the element of power set P (A)
(A) {φ {a} {b}({c} {d} {a, b} {a, c} {a, d} {b, c} {b, d} {c, d}
(B) {a, b, c} {a, b, d} {a, c, d} {b, c, d}
(C) {a, b, c, d}
(D) All the above elements are in P (A)
21. If four members a, b, c, d of a decision making body are in a meeting to pass a resolution
where rule of majority prevails list the wining coalitions. Given that a, b, c, d own 50%
20% 15% 15% shares each.
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Copyright -The Institute of Chartered Accountants of India
(A) {a, b} {a, c} {a, d} {a, b, c} {a, b, d} {a, b, c, d} (B) {b, c, d}
(C) {b, c} {b, d} {c, d} {a, c, d} {b, c, d} {a} {b} {c} {d} φ (D) None
22. As per question No.(21) with same order of options (A) (B) (C) and (D) list the blocking
conditions.
23. As per question No.(21) with same order of options (A) (B) (C) and (D) list the losing
conditions.
24. If A ={a, b, c, d, e, f} B = {a, e, i, o, u} and C = {m, n, o, p, q, r, s, t, u} then A ∪ B is
(A) {a, b, c, d, e, f, i, o, u} (B) {a, b, c, i, o, u}
(C) {d, e, f, i, o, u} (D) None
25. As per question No.(24) A ∪ C is
(A) {a, b, c, d, e, f, m, n, o, p, q, r, s, t, u} (B) {a, b, c, s, t, u}
(C) {d, e, f, p, q, r} (D) None
26. As per question No.(24) B ∪ C is
(A) {a, e, i, o, u, m, n, p, q, r, s, t} (B) {a, e, i, r, s, t}
(C) {i, o, u, p, q, r} (D) None
27. As per question No.(24) A – B is
(A) {b, c, d, f} (B) {a, e, i, o} (C) {m, n, p, q} (D) None
28. As per question No.(24) A ∩ B is
(A) {a, e} (B) {i, o} (C) {o, u} (D) None
29. As per question No.(24) B ∩ C is
(A) {a, e} (B) {i, o} (C) {o, u} (D) None
30. As per question No.(24) A ∪ (B – C) is
(A) {a, b, c, d, e, f, i} (B) {a, b, c, d, e, f, o} (C) {a, b, c, d, e, f, u} (D) None
31. As per question No.(24) A ∪ B ∪ C is
(A) {a, b, c, d, e, f, i, o, u, m, n, p, q, r, s, t} (B) {a, b, c, r, s, t}
(C) { d, e, f, n, p, q} (D) None
32. As per question No.(24) A ∩ B ∩ C is
(A) φ (B) {a, e} (C) { m, n} (D) None
33. If A = {3, 4, 5, 6} B = {3, 7, 9, 5} and C = {6, 8, 10, 12, 7} then A′ is (given that the universal
set U = {3, 4, ….., 11, 12, 13}
(A) {7, 8, ….12, 13} (B) {4, 6, 8, 10, ….13}
(C) { 3, 4, 5, 9, 11, 13} (D) None
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SETS, FUNCTIONS AND RELATIONS
34. As per question No.(33) with the same order of options (A) (B) (C) and (D) the set B′ is
35. As per question No.(33) with the same order of options (A) (B) (C) and (D) the set C′ is
36. As per question No.(33) the set (A′)′ is
(A) {3, 4, 5, 6} (B) {3, 7, 9, 5} (C) {8, 10, 11, 12, 13} (D) None
37. As per question No.(33) the set (B′)′ is
(A) {3, 4, 5, 6} (B) {3, 7, 9, 5} (C) {8, 10, 11, 12, 13} (D) None
38. As per question No.(33) the set (A ∪ B)′ is
(A) {3, 4, 5, 6} (B) {3, 7, 9, 5} (C) {8, 10, 11, 12, 13} (D) None
39. As per question No.(33) the set (A ∩ B)′ is
(A) {8, 10, 11, 12, 13} (B) {4, 6, 7, ….13} (C) {3, 4, 5, 7, 8,….13} (D) None
40. As per question No.(33) the set A′ ∪ C′ is
(A) {8, 10, 11, 12, 13} (B) {4, 6, 7, ….13} (C) {3, 4, 5, 7, 8,….13} (D) None
41. If A = {1, 2, …9}, B = {2, 4, 6, 8} C = {1, 3, 5, 7, 9}, D = {3, 4, 5} and E = {3, 5} what is set S
if it is also given that S ⊂ D and S ⊄ B
(A) {3, 5} (B) {2, 4} (C) {7, 9} (D) None
42 As per question No.(41) what is set S if it is also given that S ⊂ B and S ⊄ C
(A) {3, 5} (B) {2, 4} (C) {7, 9} (D) None
43. If U = {1, 2, …9} be the universal set A = {1, 2, 3, 4} and B = {2, 4, 6, 8} then the set A ∪ B is
(A) {1, 2, 3, 4, 6, 8} (B) {2, 4} (C) {5, 6, 7, 8, 9} (D) {5, 7, 9}
44. As per question No.(43) with the same order of options (A) (B) (C) and (D) the set A ∩ B is
45. As per question No.(43) with the same order of options (A) (B) (C) and (D) the set A′ is
46. As per question No.(43) with the same order of options (A) (B) (C) and (D) the set (A ∪ B)′ is
47. As per question No.(43) the set (A ∩ B)′ is
(A) {1, 2, 3, 4, 6, 8} (B) {2, 4} (C) {5, 6, 7, 8, 9} (D) {1, 3, 5, 6, 7, 9}
48. Let P = (1, 2, x), Q = (a x y), R = (x, y, z) then P × Q is
(A) {(1, a) (1, x) (1, y); (2, a) (2, x) (2, y); (x, a) (x, x) (x, y)}
(B) {(1, x); (1, y); (1, z); (2, x); (2, y); (2, z); (x, x) (x, y) (x, z)}
(C) {(a, x) (a, y) (a, z); (x, x) (x, y) (x, z); (y, x) (y, y) (y, z)}
(D) {(1, x) (1, y) (2, x) (2, y) (x, x) (x, y)}
49. As per question No.(48) with the same order of options (A) (B) (C) and (D) then the set
P × R is
50. As per question No.(48) with the same order of options (A) (B) (C) and (D) then the set
Q × R is
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51. As per question No.(48) with the same order of options (A) (B) (C) and (D) then the set
(P × Q) ∩ (P × R) is
52. As per question No.(48) the set (R × Q) ∩ (R × P) is
(A) {(a, x) (a, y) (a, z); (x, x) (x, y) (x, z); (y, x) (y, y) (y, z)}
(B) {(1, x) (1, y) (2, x) (2, y) (x, x) (x, y)}
(C) {(x, x) (y, x) (z, x)}
(D) {(1, a) (1, x) (1, y) (2, a) (2, x) (2, y) (x, a) (x, x) (x, y) (x, 1) (x, 2) (y, 1) (y, 2) (y, x) (z, 1)
(z, 2) (z, x)}
53. As per question No.(48) with the same order of options (A) (B) (C) and (D) as in question
No.(52) the set (P × Q) ∪ (R × P) is
54. If P has three elements Q four and R two how many elements does the Cartesian product
set P × Q × R will have
(A) 24 (B) 9 (C) 29 (D) None
55. Identify the elements of P if set Q = {1, 2, 3} and P × Q = {(4, 1) (4, 2) (4, 3) (5, 1)(5, 2) (5, 3)
(6, 1) (6, 2) (6, 3)}
(A) {3, 4, 5} (B) {4, 5, 6} (C) {5, 6, 7} (D) None
56. If A = {2, 3}, B = {4, 5}, C = {5, 6} then A × (B ∪ C) is
(A) {(2, 4) (2, 5) (2, 6) (3, 4) (3, 5) (3, 6)}
(B) {(2, 5) (3, 5)}
(C) {(2, 4) (2, 5) (3, 4) (3, 5) (4, 5) (4, 6) (5, 5) (5, 6)}
(D) None
57. As per question No.(56) with the same order of options (A) (B) (C) and (D) the set
A × (B ∩ C) is
58. As per question No.(56) with the same order of options (A) (B) (C) and (D) the set (A × B) ∪
(B × C) is
59. If A has 32 elements B has 42 elements and A ∪ B has 62 elements find the number of
elements in A ∩ B
(A) 74 (B) 62 (C) 12 (D) None
60. Out of a total population of 50000 only 28000 read Telegraph and 23000 read Times of
India while 4000 read the both. How many do not read any paper?
(A) 3000 (B) 2000 (C) 4000 (D) None
61. Out 2000 staff 48% preferred coffee 54% tea and 64% cocoa. Of the total 28% used coffee
and tea 32% tea and cocoa and 30% coffee and cocoa. Only 6% did none of these. Find the
number having all the three.
(A) 360 (B) 280 (C) 160 (D) None
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SETS, FUNCTIONS AND RELATIONS
62. As per question No.(61) with the same order of options (A) (B) (C) and (D) find the number
having tea and cocoa but not coffee.
63. As per question No.(61) with the same order of options (A) (B) (C) and (D) find the number
having only coffee.
64. Complaints about works canteen had been about Mess (M) Food (F) and Service (S). Total
complaints 173 were received as follows: –
n(M) = 110, n(F) = 55, n(S) = 67, n(M ∩ F ∩ S′) = 20, n(M ∩ S ∩ F′) = 11
and n(F ∩ S ∩ M′) = 16. Determine the complaints about all the three.
(A) 6 (B) 53 (C) 35 (D) None
65. As per question No.(64) with the same order of options (A) (B) (C) and (D) determine the
complaints about two or more than two.
66. Out of total 150 students 45 passed in Accounts 50 in Maths. 30 in Costing 30 in both
Accounts and Maths. 32 in both Maths and Costing 35 in both Accounts and Costing. 25
students passed in all the three subjects. Find the number who passed at least in any one
of the subjects.
(A) 63 (B) 53 (C) 73 (D) None
67. After qualifying out of 400 professionals, 112 joined industry, 120 started practice and
160 joined as paid assistants. There were 32, who were in both practice and service 40 in
both practice and assistantship and 20 in both industry and assistantship. There were 12
who did all the three. Find how many could not get any of these.
(A) 88 (B) 244 (C) 122 (D) None
68. As per question No.(67) with the same order of options (A) (B) (C) and (D) find how many
of them did only one of these.
69. A marketing research team interviews 100 people about their drinking habits of tea coffee
or milk or A B C respectively. Following data is obtained but the Manager is not sure
whether these are consistent.
Category No. Category No.
ABC 3 A 42
AB 7 B 17
BC 13 C 27
AC 18
(A) Inconsistent since 42 + 17 + 27 – 7 – 13 – 18 + 3 ≠ 50
(B) Consistent
(C) Cannot determine due to data insufficiency
(D) None
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70. On a survey of 100 boys it was found that 50 used white shirt 40 red and 30 blue. 20 were
habituated in using both white and red shirts 15 both red and blue shirts and 10 blue and
white shirts. Find the number of boys using all the colours.
(A) 20 (B) 25 (C) 30 (D) None
71 As per question No.(70) if 10 boys did not use any of the white red or blue colours and 20
boys used all the colours offer your comments.
(A) Inconsistent since 50 + 40 + 30 – 20 – 15 – 10 + 20 = 100
(B) Consistent
(C) cannot determine due to data insufficiency
(D) None
72. Out of 60 students 25 failed in paper (1) 24 in paper (2) 32 in paper (3) 9 in paper (1) alone
6 in paper (2) alone 5 in papers (2) and (3) and 3 in papers (1) and (2). Find how many
failed in all the three papers.
(A) 10 (B) 60 (C) 50 (D) None
73. As per question No.(72) how many passed in all the three papers?
(A) 10 (B) 60 (C) 50 (D) None
74. Asked if you will cast your vote for a party the following feed back is obtained: –
Yes No Don’t know
Adult Male 10 20 5
Adult Female 20 15 5
Youth over 18 years 10 5 10
If A = set of Adult Males C = Common set of Women and Youth Y = set of positive opinion
N = set of negative opinion then n(A′) is
(A) 25 (B) 40 (C) 20 (D) None
75. As per question No.(74) with the same order of options (A) (B) (C) and (D) the set n(A ∩ C) is
76. As per question No.(74) with the same order of options (A) (B) (C) and (D) the set n(Y ∪
N)′ is
77. As per question No.(74) with the same order of options (A) (B) (C) and (D) the set n[A ∩
(Y ∩ N)′] is
78. In a market survey you have obtained the following data which you like to examine
regarding its correctness:
Did not use the brand April May June April & May & April & April May
May June June June
Percentage answering ‘Yes’ 59 62 62 35 33 31 22
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SETS, FUNCTIONS AND RELATIONS
(A) Inconsistent since 59 + 62 + 62 – 35 – 33 – 31 + 22 ≠ 100
(B) Consistent
(C) cannot determine due to data insufficiency
(D) None
79. In his report an Inspector of an assembly line showed in respect of 100 units the following
which you are require to examine.
Defect Strength (S) Flexibility (F) Radius (R) S and F S and R F and R S F R
No. of pieces 35 40 18 7 11 12 3
(A) No. of pieces with radius defect alone was –2 which was impossible
(B) Report may be accepted
(C) Cannot be determined due to data insufficiency (D) None
80. A survey of 1000 customers revealed the following in respect of their buying habits of
different grades:
A grade A and C C grade A grade but A grade C and B None
only grades not B grade grades
180 80 480 230 360 80 140
How many buy B grade?
(A) 280 (B) 400 (C) 50 (D) None
81. As per question No.(80) with the same order of options (A) (B) (C) and (D) how many buy
C grade if and only if they do not buy B grade?
82. As per question No.(80) with the same order of options (A) (B) (C) and (D) how many buy
C and B grades but not the A grade?
83. Consider the following data: –
Skilled & Direct Unskilled & Direct Skilled & Indirect Unskilled & Indirect
Worker Worker Worker Worker
Short Term 6 8 10 20
Medium Term 7 10 16 9
Long Term 3 2 8 0
If S M L T I denote short medium long terms skilled and indirect workers respectively find
the number of workers in set M.
(A) 42 (B) 8 (C) 10 (D) 43
84. Consider the problem No.(83) and find the number of workers in set L ∩ I.
(A) 42 (B) 8 (C) 10 (D) 43
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85. Consider the problem No.(83) and find the number of workers in set S ∩ T ∩ I.
(A) 42 (B) 8 (C) 10 (D) 43
86. Consider the problem No.(83) and find the number of workers in set
(M ∪ L) ∩ (T ∪ I).
(A) 42 (B) 8 (C) 10 (D) 43
87. Consider the problem No.(83) and find the number of workers in set
S′ ∪ (S′ ∩ I)′.
(A) 42 (B) 44 (C) 43 (D) 99
88. Consider the problem No.(83). Find out which set of the pair has more workers as its
members. Pair is (S ∪ M)′ or L: –
(A) (S ∪ M)′ > L (B) (S ∪ M)′ < L (C) (S ∪ M)′ = L (D) None
89. Consider the problem No.(88). Find out which set of the pair has more workers as its
members. Pair is (I ∩ T)′ or S – (I ∩ S′): –
(A) (I ∩ T)′ > [S – (I ∩ S′)] (B) (I ∩ T)′ < [S – (I ∩ S′)]
(C) (I ∩ T)′ = [S – (I ∩ S′)] (D) None
90. Out of 1000 students 658 failed in the aggregate, 166 in the aggregate and in group-I 434
in aggregate and in group-II, 372 in group-I, 590 in group-II and 126 in both the groups.
Find out how many failed in all the three.
(A) 106 (B) 224 (C) 206 (D) 464
91. As per question No.(90) how many failed in the aggregate but not in group-II?
(A) 106 (B) 224 (C) 206 (D) 464
92. As per question No.(90) how many failed in group-I but not in the aggregate?
(A) 106 (B) 224 (C) 206 (D) 464
93. As per question No.(90) how many failed in group-II but not in group-I?
(A) 106 (B) 224 (C) 206 (D) 464
94. As per question No.(90) how many failed in aggregate or group-II but not in group-I?
(A) 206 (B) 464 (C) 628 (D) 164
95. As per question No.(90) how many failed in aggregate but not in group-I and group-II?
(A) 206 (B) 464 (C) 628 (D) 164
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SETS, FUNCTIONS AND RELATIONS
ANSWERS
1) A 2) B 3) C 4) A 5) B 6) C
7) A 8) A 9) A 10) B 11) A 12) A
13) A 14) B 15) B 16) D 17) D 18) C
19) C 20) D 21) A 22) B 23) C 24) A
25) A 26) A 27) A 28) A 29) C 30) A
31) A 32) A 33) A 34) B 35) C 36) A
37) B 38) C 39) B 40) C 41) A 42) B
43) A 44) B 45) C 46) D 47) D 48) A
49) B 50) C 51) D 52) C 53) D 54) A
55) B 56) A 57) B 58) C 59) C 60) A
61) A 62) B 63) C 64) A 65) B 66) B
67) A 68) B 69) A 70) B 71) A 72) A
73) A 74) A 75) B 76) C 77) C 78) A
79) A 80) A 81) B 82) C 83) A 84) B
85) C 86) D 87) D 88) C 89) A 90) A
91) B 92) C 93) D 94) C 95) D
77777.....3333322222 CCCCCOOOOOMMMMMMMMMMOOOOONNNNN PPPPPRRRRROOOOOFFFFFIIIIICCCCCIIIIIEEEEENNNNNCCCCCYYYYY TTTTTEEEEESSSSSTTTTT
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