2.4 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSAlgebra
- Power set :
The family of all sub-set of a given set A is known as power set and is denoted by P(A)
As for example : (i) If A = {a}, then P(A) = {{a}, φ,}
(ii) If A = {a, b}, then P(A) = {{a}, {b}, {a, b}, φ.}
(iii) If A = {a, b, c}. P (A) = {{a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}, φ.}
Thus when the number of elements of A is 1, then the number of sub-sets is 2; when the number of
elements of A is 2; then the number sub-sets is 4 = 2^2 and when it is 3, the number of sub-sets is 8 = 2^3. So,
if A has n elements, P(A) will have 2n sub-sets. - Universal Set :
In mathematical discussion, generally we consider all the sets to be sub-sets of a fixed set, known as
Universal set or Universe, denoted by U. A Universal set may be finite or infinite.
As for example :
(i) A pack of cards may be taken as universal set for a set of diamond or spade.
(ii) A set of integers is Universal set for the set of even or odd numbers. - Cardinal Number of a set :
The cardinal number of a finite set A is the number of elements of the set A. It is denoted by n{A).
e.g. : If A = {1, m, n}, B = {1, 2, 3} then n(A) = n(B)
2.1.2 Venn Diagram :
John Venn, an English logician (1834 – 1923) invented this diagram to present pictorial representation. The
diagrams display operations on sets. In a Venn diagram, we shall denote Universe U (or X) by a region
enclosed within a rectangle and any sub-set of U will be shown by circle or closed curve.
Overlapping Sets :
If two sets A and B have some elements common, these are called overlapping sets.
e.g. : If A = {2,5,7,8} and B = {5, 6, 8}, they are called overlapping sets.
Union of Sets
If A and B are two sets, then their union is the set of those elements that belong either to A or to B (or to
both).
The union of A and B is denoted symbolically as A ∪ B (read as A union B or A cup B).
In symbols, A ∪ B = {x : x ∈A or x ∈ B}
As for example :