BBA sem I Businss maths Set theory

1. Set Theory & Real Nos.
 Course: BBA
 Subject: Business Mathematics
 Unit: 1.2
1

2. Set:
 A set is a well defined collection of objects.
 The objects constituting a set are called elements or members
of the set.
 If x is an element of a set A, it is denoted by x Є A.
 If x is not an element of a set A, it is denoted by x A.
 If a set has finite number of elements it is called a finite set.
For example, A = {1,2,3} is a finite set.
 If a set has infinite number of elements it is called infinite set.
For example, A = {1,2,3………} is a infinite set.
 A set which contains no element is called a null set or an
empty set. It is denoted by Φ.


3. Examples:
1) Solve the following equations:
(a) │x-1│= 0.1 (b) │x-2│= 1
2) Express the following in the form of an interval:
(a) 0 ≤ │x-3│ ≤ 2 (b) 0 ≤ │x+5│ < 1
3) Express │3x+5│ < 2 into inequality form.

4.  Definitions:
1) Subset : If all the elements of set A are the elements of
set B (i.e. B contains all the elements of A and possibly
some more) then set A is said to be a subset of the set B.
It is symbolically written as A B.
2) Equality of two sets: Two sets A and B are said to be
equal and we write A = B if and only if A and B have the
same elements. For example, if A = {3,4,5,6} and B =
{x Є N│2< x < 7}, then A = B.
3) Universal set: All the sets will be the subset of one
fixed set is called universal set and it is denoted by U.


5. 4) Singleton set: A set containing only one element is called
a singleton set. For example: A = {5} is a singleton set.
5) Disjoint set: Two sets are said to be disjoint if they have
no element in common. For example, A = {1,2,3} and
B = {6,7,8} are disjoint sets.
6) Power set: A set consisting of all subsets of a given set is
called its power set. The power set of a set A is denoted by
P(A). For example, if A= {a,b,c} then
P(A) = {Φ, a, b, c, {a,b}, {a,c}, {b,c}, A}

6. Set Operations:
1) Intersection of two sets: The intersection of two sets A
and B is the set of all elements which belong to both A
and B. It is denoted by A ∩ B.
Thus, A ∩ B {x│ x Є A and x Є B }.
for example, A = {1,2,3,4}; B = {2,4}, A ∩ B = {2,4}
2) Union of two sets: The union of two sets A and B is the
set of all elements which belong to A or B or to both. It
is denoted by A U B.
Thus, A U B {x│ x Є A and / or x Є B }.
for example, A = {1,2,3,4,5}; B = {2,4,6}, then
A U B = {1,2,3,4,5,6}

7. 3) Complement of a set: Let U be the universal set and A be
a subset of U, then the complement of set A is another
subset containing all the elements of U which are not in A.
It is denoted by A’ or .
Thus A’ = {x│ x A and x Є U }.
4) Difference of two sets: The difference A-B of two sets A
and B is the set of all elements of A which are not element
of B.
Thus A-B = {x│ x Є A and x B }.
5) Symmetric Difference: The set of all elements which
are in A or in B but not in both is called the symmetric
difference set of sets A and B. It is denoted by A ∆ B.
Thus, A ∆ B = (A U B) – (A ∩ B ).
A



8. Examples:
1) If A = {1,2,5} and B = {n │ n Є N}, then find A U B &
A ∩ B .
2) Let the universal set be U = {x │3 ≤ x ≤ 13, x Є N},
A = {y │2 < y < 7, y Є N} and B = {3,5,7,9}. Then find
(i) A’ (ii) (B’)’ (iii) (AUB)’
3) If U = {x │1 ≤ x ≤ 10, x Є N}, A = {1,2,3,4},
B = {2,4,6,8} and C = {3,4,5,6} then find (i) AUB (ii)
AUC (iii) BUC (iv) C ∩ A (v) A ∩ B (vi) C ∩ B (vii) B-
C (viii) (AUB)’ (ix) A’, B’ C’ (x) A-B (xi) A-C (xii) C-A
(xiii) A ∆ C

9. 4) State the associative and distributive laws for three sets
A, B, C and verify them by taking A = {1,2,5,6,8} ,
B = {2,4,6,10,11} and C = {1,2,3,5,6,11,12}.
5) State the De Morgan’s law for two sets A and B and
verify them by taking U = {1,2,3,4,5,6,7,8,9,10},
A = {1,2,4,6,8} and B = {2,3,6,7,9}.
6) If U = {p,q,r,s}, A = {p,q,r} and B = {q,r,s}, verify that
(AUB) – B = A ∩ B’
7) If A = {1,2,3,4,5,6} and B = {2,5,6,7,8,9,10} then find
A ∆ B.
8) If A={x │2 < x < 9, x Є N}, & B ={x │x2 <5x , x Є N}
then find A ∆ B.

10. Cartesian Product of two sets:
Let A and B be two given non empty sets.
The set of all ordered pairs (x, y) where x Є A and y Є B
is called the Cartesian product of sets A and B.
It is denoted by A × B.
Thus, A × B = {(x, y) │ x Є A and y Є B }
B × A = {(y, x) │ y Є B and x Є A }.
For example: if A = {1,2,3} and B = {5,6} then
A × B = {(1,5), (1,6), (2,5), (2,6), (3,5), (3,6)} &
B × A = {(5,1), (5,2), (5,3), (6,1), (6,2), (6,3)}.

11. 9) If U = set of letters of the word ‘WHEAT’, A = set of
letters of the word ‘WHAT’, B = set of letters of the word
‘HEAT’ , C= set of letters of the word ‘EATA’, then find
(i) (A ∩ B) × (B ∩ C ) (ii) A ∩ (B - C)
(iii) (A - B)’ ∩ C’ (iv) (A ∩ B ∩ C)’
10) If A = {5,6,7}, B = { 7,8} and C = {5,8} then verify the
following results:
(i) A × (B - C) = (A × B) – (A × C)
(ii) A × (B ∆ C) = (A × B) ∆ (A × C).
(iii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iv) A × (B U C) = (A × B) U (A × C).
11) If n(A ×A) = 16, (1,2) Є A ×A, (3,4) Є A ×A, then write
all members of A and A ×A.

12. 2. 3.
4. 5.

13. 6.
7.
8.
9.
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14. Examples:
1) A town has a total population of 50,000 persons and of them 28,000
read ‘Gujarat Samachar’ and 23,000 read ‘Gujarat Mitra’, while 4,000
read both the papers. Indicate how many read neither ‘Gujarat
Samachar’ nor ‘Gujarat Mitra’.
2) A class of 100 students appeared for F.Y.B.B.A. examination . Out of
100 students, 40 passed in mathematics, 36 passed in management, 60
passed in accountancy, 8 students passed in mathematics and
management, 17 passed in management and accountancy, 16 passed in
mathematics and accountancy and 5 passed in all the three subjects.
Find (i) how many students passed in exactly one subject and (ii) how
many students passed in at least two subjects.
3) There are 60 students in a class of a college. 15 students do not have
interest in games and they do not take part in the games. 32 students of
the class play football and 18 students play hockey. Then find (i) how
many students play both the games? (ii) how many students play only
football? (iii) how many students play only hockey?

15. References:
1.http://mathforum.org/mathimages/imgUpload/Venn_
Diagram.jpg
2. https://lh5.ggpht.com/TLC-l_QkYo5Ka-Ycj-ikrRS-g71-
J3AtDkpyc2fTZM6Xa3lsWvsXRIIgyh_BA-5-
Hs8CiQ=s170
3.https://lh3.ggpht.com/Gjpj82W8fh2gdeL4YhYOKyyRH
g1b0xUk15jaBTNdT_vP8HNsx98pKmMDNiSlYaErDEqq
=s135
4.https://lh6.ggpht.com/95wixCkKu5PVLsXfT9ssKpNiH
KkepUfwkBXSpRQJjLHxS9zXt6oQjbfaWrsUwHix7Nxb
=s170

16. 5.https://lh6.ggpht.com/blUTj3GXCQUc6ij4CKHZFuVXpyQ5v_
hLY5adQfLsdJTArOQdFlnKKtu_uPBCw5ONFV5=s135
6.https://lh4.ggpht.com/Siz8CO3dX2lfSsEpGOub_iywASUucFU
1eToZbuQUQP4yLSzlw3F4irzNwrNi44U2r_zw=s149
7.https://lh3.ggpht.com/xkRj_3ny_yMUimF0beyOYgiGyAf_8VlL
BgwNrTKVh7yT2filVMbjksPabQh4TQtzksFXA8=s113
8. https://lh5.ggpht.com/5m9uEDgxI-bOkU-6Qf2-
6QEPckm7pEoazFGsRy0mLlya4AtyiMERY7HCyUGiw7GO9x1
dOes=s139
9.https://lh6.ggpht.com/ZpIEjkHi8QoVEJmFeK9YpY12rC7k4zD
7PMZbIybrmhJmboYNasfg___OSOjPFoWdl4wUeQ=s126
10.https://lh6.ggpht.com/Yjyl7W2TY70d5NbcCQzr8un7Ymwxd
hQD27wpVT35VDvhleoQuTLKYy2Cu3JHFnzP9crRbw=s106

17. 1. Business Mathematics by A. G.Patel and G.C. Patel –
Atul Prakashan. (for content)