Chapter 1 | Fundamentals of Mathematics
Chapter
1
28
Fundamentals of
Mathematics
LECTURE-5 SET THEORY
DEFINITION
A set is a collection of well-defined objects which
are distinct from each other. Sets are generally
denoted by capital letters A, B, C, ....... etc. and the
elements of the set by small letters a, b, c ....... etc.
If a is an element of a set A, then we write a  A
and say a belongs to A.
If a does not belong to A then we write a A,
e.g. the collection of first five prime natural numbers
is a set containing the elements 2, 3, 5, 7, 11.
NOTATION
1.
2.
3.
4.
5.
6.
7.
8.
Suppose S is a set (i.e., a collection of objects
enjoying a certain property).
The symbol  stands for “belongs to” or “is a
member of”: a  S means a belongs to S or is a
member of S.
The symbol  stands for “does not belong to”. a 
S means a is not a member of S.
The symbols  and  stand for “there exists” and
“such that”, respectively. Generally, these two
symbols go together, for example,  real number x
 x2 = 2.
The symbol ““ stands for “for all” or “for every”
For example, “x2 is a positive integer  non-zero
integer x”
If S is the set of all objects satisfying a property P,
then S is represented as
S = {x | x has property P}
The set having no objects is called the empty set
or null set and is denoted by ““.
If a set has only a finite number of members x1, x2,
…., xn, then we write
S = {x1, x2, …., xn}
The symbol  is read as “implies”. Thus, a  b is
read as “a implies b”. The symbol  is read as
“implies and is implied by” or as “if and only if”.
Thus, a  b is read as “a implies and implied by b”
or “an if and only if b”.
Z0 = The set of all non-zero integers
= {±1, ±2, ±3, ....}
Q = The set of all rational numbers
p
q

=  : p,q  I,q  0 

R = The set of all real numbers
R – Q = The set of all irrational numbers
e.g. 2, 3, 5 ,...., e, log2 etc. are all irrational
numbers.
METHODS TO WRITE A SET
(i) Roster Method or Tabular Method : In this method a
set is described by listing elements, separated by
commas and enclose then by curly brackets. Note
that while writing the set-in roster form, an element
is not generally repeated e.g. the set of letters of
word SCHOOL may be written as {S, C, H, 0, L}.
(ii) Set builder form (Property Method) : In this we
write down a property or rule which gives us all
the element of the set.
A = {x : P(x)} where P(x) is the property by which
x  A and colon ( : ) stands for ‘such that’.
Example 5.1
Express set A = {x : x  N and x = 2n for n  N} in
roster form
Solution : A= {2, 4, 8, 16, .........}
Example 5.2
Express set B = {x3 : x < 5, x  W} in roster form
Solution : B = {0, 1, 8, 27, 64}
Example 5.3
Express set A = {0, 7, 26, 63, 124} in set builder form
Solution : A = {x : x = n3 – 1, n  N, 1 ≤ n ≤ 5}
SOME IMPORTANT NUMBER SETS
N
= Set of all-natural numbers
= {1,2,3,4, ....}
W = Set of all whole numbers
= {0, 1, 2, 3, ....}
Z or I set of all integers
= {.... – 3, – 2, – 1,0, 1,2,3, ....}
Z+ = Set of all +ve integers
= {1,2,3, ....} = N.
Z– = Set of all –ve integers
= {–1, –2, –3, ....}
(iii) Arrow-diagram form : In this form, we draw an
arrow corresponding to each ordered pair (a, b)
in R from the first component a to the second
component b. For example, consider the relation
R given as below. Then R can be represented as
shown in Figure. There are nine arrows
corresponding to nine ordered pairs belonging to
the relation R.
R = {(2, 2), (2, 4), (2, 6), (2, 8), (2, 10), (3, 6), (4,
4), (4, 8), (5, 10)}
By : Prashant Jain Sir
29
Chapter 1 | Fundamentals of Mathematics
Universal set : A set consisting of all possible
elements which occur in the discussion is called a
universal set and is denoted by U
Note
All sets are contained in the universal set.
Ex. If A= {1, 2, 3} B = {2, 4, 5, 6}, C = {1, 3, 5, 7},
then = {1, 2, 3, 4, 5, 6, 7} can be taken as the
universal set.
Power set : Let A be any set. The set of all subset of A
is called power set of A and is denoted by P(A).
Ex. 1 Let A = {1, 2} then P(A) = {, {1}, {2}, {1, 2}}
Ex. 2 Let P() = {}
 P(P()) = {, {}}
 P(P(P( )) = {, {}, {{}}, { { }}
FIGURE : Representation of arrow-diagram form.
TYPES OF SETS
Null set or Empty set : A set having no element in
it is called an Empty set or a null set or a void set
it is denoted by  or { }
Ex. A = (x  N : 5 < x < 6} = 
A set consisting of at least one element is called a
non-empty set or a non-void set.
Singleton : A set consisting of a single element is
called a singleton set.
Ex. Then set {0}, is a singleton set
Finite Set : A set which has, only finite number of
elements is called a finite set.
Ex. A= {a, b, c}
Order of a finite set : The number of element in a
finite set A is called the order of the set A and is
denoted O(A) or n(A). It is also called cardinal
number of the set.
Ex. A = {a, b, c, d}  n(A) = 4
Infinite set : A set which has an infinite number of
elements is called an infinite set.
Ex. A = {l, 2, 3, 4, .....} is an infinite set
Equal sets : Two sets A and B are said to b equal if
every element of A is a member of B, and every
element of B is a member of A.
If sets A and B are equal. We write A = B and A and
B are not equal then A  B
Ex. A = {l, 2, 6, 7} and B = {6, 1, 2, 7}  A = B
Equivalent sets : Two finite sets A and B are
equivalent if their number of elements are same
i.e. n(A) = n(B)
Ex. A = {1, 2, 3, 7} B = {a, b, c, d}
n(A) = 4 and n(B) = 4  n(A) = n(B)
Note
Equal set always equivalent but equivalent sets
may not be equal
Subsets : Let A and B be two sets if every element
of A is an element of B, then A is called a subset of
B if A is a subset of B. we write A  B
Ex. A = {l, 2, 4} and B = {1, 2, 3, 4, 5, 6, 7}  A  B
The symbol “” stands for “implies”
Proper subset : if A is a subset of B and A  B then
A is a proper subset of B. and we write A  B.
1.
2.
3.
4.
Note
Every set is a subset of itself i.e. A  A for all A
Empty set , is a subset of every set
Clearly N  W  Z  Q  R  C
The total number of subsets of a finite set
containing n elements is 2n
By : Prashant Jain Sir
1.
2.
Note
If A= then P(A) has one element
Power set of as given set is always non empty
Some Operation on Sets :
(i) Union of two sets : A  B = {x : x  A or x  B}
e.g. A = {1, 2, 3} B = {2, 3, 4} then A  B = {1, 2, 3 4}
(ii) Intersection of two sets : A  B = {x : x  and x  B}
e.g. A = {1, 2, 3} B = {2, 3, 4} then A  B = {2, 3}
(iii) Difference of two sets : A – B = {x : x  A and x  B}
e.g. A = {1, 2, 3} B = {2, 3, 4}; A – B = {1}
(iv) Complement of a set : A’ = {x : x  A but x  U}
=U–A
e. g. U = {1, 2, ….., 10} A = {1, 2, 3, 4, 5} then
A’ = {6, 7, 8, 9, 10}
(v) De-Morgan Laws : (A  B)’ = A’  B’;
(A  B)’ = A’  B’
(vi) A – (B  C) = (A – B)  (A – C);
A – (B  C) = (A – B)  (A – C)
(vii) Distributive Laws : A  (B  C) = (A  B)  (A  C);
A  (B  C) = (A  B)  (A  C)
(viii) Commutative Laws : A  B = B  A; A  B = B  A
(ix) Associative Laws : (A  B)  C = A  (B  C);
(A  B)  C = A  (B  C)
(x) A   = ; A  U = A
A   = ; A  U = U
(xi) A  B  A; A  B  B
(xii) A  A  B; B  A  B
(xiii) A  B  A  B = A
(xiv) A  B  A  B = B
Disjoint Sets : If A  B = . then A, B are disjoint.
e.g. if A = {1, 2, 3} B = {7, 8, 9} then A  B = 
Note
A, A’ are disjoint if A  A’ = 
Symmetric Difference of sets :
A  B = (A – B)  (B – A)
• (A’)’ = A
• A  B  B’  A’
If A and B are any two sets, then
(i) A – B = A  B’
Chapter 1 | Fundamentals of Mathematics
(ii) B – A = B  A’
(iii) A – B = A  A  B = 
(iv) (A – B)  B = A  B
30
(v) (A – B)  B = 
(vi) (A – B)  (B – A) = (A  B) – (A  B)
WORKED-OUT PROBLEMS - 5
1.
Verify whether the following are sets:
(1) The collection of all intelligent persons in
Visakha-patnam.
(2) The collection of all prime ministers of India.
(3) The collection of all negative integers.
(4) The collection of all tall persons in India.
Solution : Note that the collections given in (1) and (4)
are not sets because, if we select a person in
Visakhapatnam, we cannot say with certainty whether
he/she belongs to the collection or not, as there is no
stand and scale for the evaluation of intelligence or for
being tall. However, the collections given in (2) and
(3) are sets.
(iv) D = {2, 3, 5}
(v) E = {T, R, I, G, O, N, M, E, Y}
(vi) F = {B, E, T, R}
2.
6.
Which of the following are sets? Justify your
answer.
(i) The collection of all the months of a year
beginning with the letter J.
(ii) The collection of ten most talented writers of
India.
(iii) A team of eleven best-cricket batsmen of the
world.
(iv) The collection of all boys in your class.
(v) The collection of all-natural numbers less
than 100.
(vi) A collection of novels written by the writer
Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the
world.
Solution : (i), (iv), (v), (vi), (vii) and (viii) are sets.
3.
(1) Let P be the collection of all prime numbers.
Then it can be represented in the set builder
form as
(2) Let X be the set of all even positive integers
which are less than 15. Then express this set
in set builder and roster form.
Solution : (1) P = {x|x is a prime number}
(2) X = {x|x is even integer and 0 < x < 15}
= {2,4,6,8,10,12,14}
4.
Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such
that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor
of 60}
(v) E = The set of all letters in the word
TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
Solution : (i) A = {–3, –2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}
5.
Write the following sets in the set-builder form :
(i) {3, 6, 9, 12}
(ii) {2,4,8,16,32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, . . .}
(v) {1,4,9, . . .,100}
Solution : (i) {x : x = 3n, n  N and 1 ≤ n ≤ 4}
(ii) {x : x = 2n, n  N and 1 ≤ n ≤ 5}
(iii) {x : x = 5n, n  N and 1 ≤ n ≤ 4}
(iv) {x : x is an even natural number}
(v) {x : x = n2, n  N and 1 ≤ n ≤ 10}
Fill in the blank infinite/finite
(1) The set Z+ of positive integers is an ______ set.
(2) {a, b, c, d} is a _____ set, since it has exactly four
elements.
(3) The set R of real numbers is an _________ set.
(4) {x|x  Z and 0 < x ≤ 100} is _________ set.
(5) {x|x  Q and 0 < x < 1} is an _________ set.
Solution : (1) Infinite
(2) Finite
(3) Infinite
(4) Finite
(5) Infinite
7.
Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate
symbol  or  in the blank spaces:
(i) 5. . .A
(ii) 8 . . . A
(iii) 0. . .A
(iv) 4. . . A
(v) 2. . .A
(vi) 10. . .A
Solution : (i) 
(ii) 
(iii) 
(vi) 
(v)  (vi) 
8.
In the following, state whether A = B or not:
(i) A = {a, b, c, d} B = {d, c, b, a}
(ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10} B = {x : x is positive even
integer and x ≤ 10}
(iv) A = {x : x is a multiple of 10}, B = {10, 15, 20,
25, 30, . . . }
Solution : (i) Yes (ii) No
(iii) Yes (iv) No
If  denotes null set then find
(a) P()
(b) P(P())
(c) n(P(P(P())))
(d) n(P(P(P(P()))))
Solution :
(a) P() = {}
(b) P(P()) = {,{}}
(c) n(P(P(P()))) = 22 = 4
(d) n(P(P(P(P())))) = 24 = 16
9.
10. State true/false :
(1) A = {p, q, r, s}, B = {p, q, r, p, t} then A  B.
(2) A= {p, q, r, s}, B = {s, r, q, p} then A  B.
(3) [4, 15)  [–15, 15]
Solution :
By : Prashant Jain Sir
31
Chapter 1 | Fundamentals of Mathematics
(1) False
(2) True
(3) True
11. Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x2 + 5x + 6 = 0}
(ii) A = {x : x is a letter in the word FOLLOW}
B = {y : y is a letter in the word WOLF}
Solution : (i) No (ii) Yes
(2) A= {a, b, c, d, e) and B = (b, c, f, g)
Solution :
(1) We have A = {1, 2, 3, 4} and B = {4, 5, 6}. Then
A – B = {1, 2, 3} and B – A = {5, 6}
Therefore
A  B = {1, 2, 3}  {5, 6} = {1, 2, 3, 5, 6}
(2) From the given sets we have
A – B = {a, d, e} and B – A = {f, g}
Therefore
A  B = {a, d, e}  {f, g} = {a, d, e, f, g)
12. From the sets given below, select equal sets :
A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14},
D = {3, 1, 4, 2}, E = {–1, 1}, F = { 0, a}, G = {1, –1},
H = {0, 1}
Solution : B = D, E = G
18. Let A = The set of all non-negative integers and B =
The set of all non-positive integers. Find A  B.
Solution : A  B = {x|x is an integer, x ≥ 0 and x ≤ 0} = {0}.
13. Let A= {x|x is an odd prime and x < 20} and B =
{x|x is an integer and x > 6}. Find A  B.
Solution : A = {3, 5, 7, 11, 13, 17, 19} and
B = {7, 8, 9, 10, 11, 12, ...}
Therefore,
A  B = {7, 11, 13, 17, 19}
19. The set A = {x : x  R, x2 = 16 and 2x = 6} is equal to(l) 
(2) {14, 3, 4}
(3) {3}
(4) {4}
Solution : x2 = 16  x = ±4
2x = 6  x = 3
There is no value of x which satisfies both the
above equations.
Thus, A = 
Ans. (1)
14. Let A be the interval [0, 1] and B the interval [1/2, 2].
Then find A  B and A  B.
Solution : We have
A  B = {x|x  A or x  B}
1
= {x|x  R and ‘0≤ x ≤ 1 or
≤ x ≤ 2’}
2
= {x|x  R and 0 ≤ x ≤ 2}
= [0, 2]
Also,
1 
A  B =  ,1
2 
15. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13,
15} and D = {15, 17}; find
(i) A  B
(ii) B  C (iii) A  C  D
(iv) A  C
(v) B  D (vi) A  (B  C)
(vii) A  D
(viii) A  (B  D)
(ix) (A  B)  (B  C) (x) (A  D)  (B  C)
Solution : (i) {7, 9, 11}
(ii) {11, 13}
(iii) 
(iv) {11}
(v) 
(vi) {7, 9, 11}
(vii) 
(viii) {7, 9, 11}
(ix) {7, 9, 11}
(x) {7, 9, 11}
16. Let A be the set of all even primes and B the
interval (2, 3). Find A  B.
Solution : A  B = {x|x is an even prime or x  (2, 3)}
= {x|x = 2 or x  R. such that 2 < x < 3}
= {x|x  R. and 2 ≤ x < 3}
= [2, 3)
17. Find the symmetric difference of the following:
(1) A= {1, 2, 3, 4} and B = {4, 5, 6}
By : Prashant Jain Sir
20. Let A = {x : x  R, |x| < 1}; B = {x : x  R, |x – 1| ≥
1} and A  B = R – D, then the set D is(l) {x : 1< x ≤ 2}
(2) {x : 1 ≤ x < 2}
(3) {x : 1 ≤ x ≤ 2}
(4) none of these
Solution :
A = {x : x  R, –1 < x < 1}
B = {x : x  R : x – 1 ≤ –1 or x – 1 ≥ 1}
= {x : x  R : x ≤ 0 or x ≤ 2}
AB=R–D
where D = {x : x  R, 1 ≤ x < 2}
Ans. (2)
21. If aN= {ax : x  N}, then the set 6N  8N is equal
to(1) 8N
(2) 48N (3) 12N (4) 24N
Solution : 6N = {6, 12, 18, 24, 30, .....}
8N = {8, 16, 24, 32 .....}

6N  8N = {24, 48, ......} = 24N
Short cut Method
6N  8N=24N [24 is the L.C.M. of 6 and 8]. Ans. (4)
22. If R is relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8,
9} given by x Ry  y = 3x then R =
(A) {(3, 1), (6, 2), (8, 2), (9, 3)}
(B) {(3, 1),
(6, 2), (9, 3)}
(C) {(3, 1), (2, 6), (3, 9)}
(D) {(1, 3),
(2, 6), (3, 9)}
Solution : x = 1  y = 3
x=2y=6
x = 3  y = 9  R = {(1, 3), (2, 6), (3, 9)}