Mathematical Preliminaries
Mathematical Preliminaries
Topics
• Set Theory
• Combinatorics
• Mathematical Functions
• Summations
• Probability Theory
Set Theory
Set Theory
Definitions
A Set is a collection of objects called members. If a, b, c, d are members of a set S we write :
S ={a, b, c, d}
where a, b, c, d
S ( read a, b, c, d belong to S )
The set builder notation can be used to define a set. For example, the set
S={1, 3, 5, 7, 11, 13}
can be defined as
S = {x: where x is prime number and x<15}
In set notation the order of members does not matter
S = {a, b, c, d} is identical to set {d, b, c, a}
In a set all members are distinct objects , i. e, duplicate members are not allowed
A set that contains no elements is called empty set. It is symbolically denoted by φ
Set Theory
Subsets
A set A is called a subset of set B if every element of A is also a member B. Symbolically:
A B, if x A then x B
The set A is proper subset of B, if A contains only some of the elements of B. Symbolically
the relationship is represented as:
A B
The relationship among sets is usually shown by a picture, which is called Venn diagram.
Figures (a), (b) show Venn diagrams for a subset and proper subset
(a) A is proper subset of B (b) A is subset of B
B
A
A B
A=B
A

B
Set Theory
Universal Set
The largest set for a given collection of objects is called Universal Set. All other sets are
subsets of the Universal set U. Symbolically:
φ U, A
U, BU, C U
Example: The Universal set, U, of all small English alphabets is
U = {a, b, c,…..,x, y, z}
The subsets of the Universal set are, for example, A = {a, f, g}, B = {p, q, r, s , t},
C = {x ,y, w}, as shown in the diagram
Given a Universal set, we define complement Ac of set A as
Ac = U – A
The set Ac contains all of elements of Universal set, excluding those in the set A.
Example: If A={a, b, c}, then Ac = { d, e, f,……, x, y, z}
Subsets of Universal set of alphabets (a, b, c,…x, y ,z}
C
B
U
Set Operations
A
B
A
B
U
n
i
o
n
s
e
t
Union
The union of set A and set B is set of all elements x such that x is in A or x is in B.
Symbolically:
A U B, if x A or x B
The Union operation is pictorially represented by Venn diagram, as shown below:
Example (1): A={ 1, 3, 4, 5}, B={ 3, 4, 7, 8, 9}. The Union of sets A and B is the