Set Theory
Sets
A set is a well-defined collection of objects called elements or members of the set.
Well-defined means it must be possible to unambiguously decide whether a given object
belongs to the collection or not.
Examples: A = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
B = {1, 2, 3}; C = {x : x is a whole number and 1 < x < 5}
Uppercase letters A, B, C etc are used to denote letters and lowercase letters a, b, c … to
denote the members or elements of sets.
Defining a Set
A set may be defined in two ways:
1. By listing all the elements between braces { }. This method is called Roster
method. Example: A = {1, 2, 3, 4}. We say, 1  A; 2  A and so on. But 7  A.
2. Give a rule by which all its members can be found (Set-builder method). For
example, Z = {x | x is an integer}
In the roster method, the order in which the elements of a set are listed is not important.
Further, repeated elements are listed only once.
Specific Sets
C
R
R+
R–
Z
Z+
Z–
N
Q

The set of complex numbers
The set of all real numbers
The set of all positive real numbers
The set of all negative real numbers
The set of all integers. Hence Z = { …, –3, –2, –1, 0, 1, 2, 3, …}
The set of all positive integers. Hence Z+ = {1, 2, 3, …}
The set of all negative integers
The set of natural nunbers
The set of all rational numbers, i.e. number of the form p/q, p  Z; q  N
The empty set, i.e. the set with no elements. Thus  = { }
Equal Sets
Two sets A and B are equal if every element of either set is also an element of the other.
We can then write A = B.
E.g.: A = {1, 2, 3} and B = {x : x is a positive integer and x2 < 12}.
Subsets
If every element of set A is also an element of set B, A is said to be a subset of B.
We write:
AB
If the set B contains at least one more element then we say A is a proper subset of B and
write:
AB
Examples;
Z  Q;
RC
Notes:
1. Every ser is a subset of itself. Thus if A is any set, A  A
2. The null set  is a subset of every set because there is no element in  that are not
in any given set
3. A = B  A B and B  A
Finite and Infinite sets
Set A is said to be finite if it has n distinct elements, where n  N. The number of
elements, n, is called the cardinality of A and is denoted by n(A) or |A|.
A set that is not finite is called an infinite set.
Power Set
The set of all subsets of a set A is called the power set of A and is denoted by P(A). The
number of elements in P(A) is 2n, where n is the number of elements in A.
Example: If A = {1, 2, 3} then P(A) = {, {1}, {2}, {3}, {1,2}, {2,3}, {3,1}, (1,2,3}}.
Venn Diagrams
A graphical representation of sets and their relationships is possible by Venn diagrams.
Here a set is represented by a closed figure, usually a circle, which encloses all the
elements of that set. The closed figure is drawn inside a rectangle that represents the
universal set, X or U, which contains all items of interest in a given context.
Operations on Sets
Union: If A and B are two sets, we define the union of A and B as the set containing all
elements that belong to A or B.
Thus, ‘A union B’ is A  B = { x | x  A or x  B or both}
Intersection: If A and B are two sets, we define their intersection as the set containing all
elements that are common to A and B.
Thus, ‘A intersection B’ is A  B = { x | x  A and x  B}
If AB = , i.e. A and B have no common element, then they are said to be disjoint sets.
Difference of Sets
The difference A – B is the set of all elements that belong to A, but not to B.
A – B = { x | x  A and x  B}
Complement
If A is a set and X the appropriate universal set, then the complement of A, represented
by A’ orA, is the set of all elements in X that are not members of A.
Thus: ‘A complement’ is A = X – A = { x | x  A}
Symmetric Difference
If A and B are two sets, their symmetric difference is the set of elements that belong to A
or B, but not to both.
Thus A  B = (A – B)  (B–A) = { x | (x  A and x  B) or (x  B and x  A)}
Set Algebra
Operations on sets satisfy the following properties:
1. Commutative properties: AB = BA
and
AB = BA
2. Associative properties: A(BC) = (AB)C and
A(BC) = (AB)C
3. Distributive properties: A(BC) = (AB) (AC); A(BC) = (AB) (AC)
4. Complement laws:
A A = X
and
A A = 
5. Identity laws:
AX =A
and
A=A
6. Idempotent laws:
AA=A
and
AA=A
7. De Morgan’s laws:
(AB)’ = A’B’
and
(AB)’ = A’B’
8. Involution law:
(A’)’ = A
Principle of duality: If in any law, we interchange  and , as well as X and , we get
another valid law. This is referred to as the principle of duality.
The addition principle or the inclusion-exclusion principle
If A, B, C are finite sets, then
1. n(AB) = n(A) + n(B) – n(AB)
2. n(ABC) = n(A) + n(B) + n(C) – n(AB) – n(BC) – n(CA) + n(ABC)
Example:
1. If A = {a, b, c, d, e}; B = {a, b, e, g, h} and C = {b, d, e, g, h, k, m, n}, verify the
addition principle.
2. A computer company wants to hire 25 programmers to handle systems programming
jobs and 40 programmers for application programming. Of those hired, 10 will be
expected to perform both jobs. How many programmers must be hired? Ans: 55.
3. A survey was undertaken on the means of commuting. Each respondent was asked to
check bus, train or car as the major method of travelling to work. More than one
answer was permitted. The results were as follows: bus - 30 persons; train - 35; car 100; bus and train - 15; bus and car - 15; train and car - 20; and all the three - 5
persons. How many persons participated in the survey? Ans: 30+35+100–15–15–
20+5 = 120.
4. An analysis of 100 faults occurring on a production line showed that in 30 of the
cases, the fault was due to faulty material, 15 of the cases to incorrect use of a tool
and 70 of the cases to poor workmanship. If there were 19 cases involving two of
these faults, how many involved all the three? Ans: 100 = 30+15+70–19+x  x = 4.
5. Quality control on a production line checks each product against three tests. When
checking 400 items it was found that 39 failed test A, 30 failed test B and 28 failed
test C. Of these, 7 failed tests A & B, 6 failed tests B & C and 5 failed tests A & C. 3
failed all the tests. Determine the number of items that failed just one test. Ans: 70.
How many items failed at least one test? Ans: 39+30+28–7–6–5+3=82.
Product Sets
If A and B are two non-empty sets, the product set or Cartesian product AXB is the set of
all ordered pairs (a, b) where a  A and b  B. Thus
AXB = {(a,b) | a  A and b  B}
Examples
1. If A = {Kris, Bob, Kichu} and B = {Alice, Aparajita} then find AXB and BXA.
2. If A = {a | a is a real number} and B = {1, 2, 3}, sketch AXB and BXA in the
Cartesian plane. Ans: To sketch AXB, take A along the x-axis and B along the yaxis. The set AXB is represented by three straight lines y = 1, y = 2 and y = 3. To
sketch BXA, take B along the x-axis and A along the y-axis. The set BXA is
represented by the three straight lines x = 1, x = 2 and x = 3.