business mathematics

Cartesian Product:

Let A and B be any two sets. Then the Cartesian product of A and B is the set of all ordered pairs of the form (a, b), where aЄA and bЄB
The product is denoted by A×B
A×B = {(a, b)/ aЄA, bЄB }

Example
A = {a, b, c} and B={1,2}, then
A×B = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2)}
B×A = {(1, a), (2, a), (1, b), (2, b), (1, c), (2, c)}

Null Set:
A set is said to be a null set if it has no elements. It is also called an empty set or a void set and is denoted by ø.
Example:
{x | x is a perfect square and 4< x <9}

Finite and infinite sets:
A set ‘A’ is said to be finite if it is either an empty set or contains finite number of elements. Otherwise it is called an infinite set.
Example:
Set of natural numbers less than 100 is finite.

Cardinality of a finite set:
The number of distinct elements in a set is called the cardinality of the set. If a finite set A has n distinct elements, the cardinality of the set is n and is denoted by O(A) or n(A). The cardinality of an empty set is zero.
Example:
Cardinality of A = {a,e,I,o,u} is 5.

A set is a well defined collection objects. The objects of the set are called its elements. Sets are usually denoted by capital letters and the elements of the set are denoted by lower case. If an element x belongs to set A, it is denoted by x Є A. If x is not an element of A, it is denoted by striped(/)Є A.

A set, in general is represented in two forms:

1) In this form, a set is described by actually listing out all the elements. For example, the set of all odd natural numbers less than 10 is represented by {1,3,5,7,9}.

2) In this form, a set is described by a characterizing property. For example, the set of all odd natural numbers less than 10 is represented by {x Є N | x < 10 and x is odd}. The symbol | is read as “ such that.”

1) A ⋃ A = A; A ⋂ A = A
2) A ⋃ B = B ⋃ A
3) A ⋂ B = B ⋂ A
4) A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ C
5) A ⋂ (B ⋂ C) = (A ⋂ B) ⋂ C
6) A ⋃ (B ⋂ C) = (A ⋃ B) ⋂(A ⋃ C)
7) A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)
8) C - (C - A) = C ⋂ A
9) C - (A ⋂ B) = (C - A) ⋃ (C - B)
10) C - (A ⋃ B) = (C - A) ⋂ (C - B)

Union of sets:
If A and B are two sets, the union of A and B is the set of all those elements which belong to either A or B or both A and B and is denoted by A ⋃ B.
A ⋃ B = {x| x Є A or x Є B}
If A ⊂ B, then A ⋃ B = B
A ⋃ Φ = A
A ⋃ μ = μ

Intersection of Sets:
Let A and B, be two sets. The intersection of A and B is the set of all those elements that belong to A and B. It is denoted by A ⋂ B.
A ⋂ B = {x | x Є A and x Є B}
If A ⊂ B, then A ⋂ B = A
A ⋂ Φ = Φ
A ⋂ μ = A

Difference of Sets:
Let A and B be two sets. The difference of A and B, denoted by A - B, is the set of all those elements of A which do not belong to B.
A – B = {x| x Є A and x∉ B} = { x Є A | x ∉ B}
Similarly, B – A = { x Є B | x ∉A}
Example:
A = {4,5,6,7}, B = {6,7,8,10}
A – B = {4,5}
B – A = {8,10}