MDU BBA NOTES
SEMESTER-1
Business Mathematics
Meaning of Sets, Elements, And Types.
Meaning of Sets
The concept of set serves as a fundamental part of present-day mathematics. Today this concept is being used in almost every branch of mathematics. Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.
Meaning of Sets – A set is a collection of objects called Sets. Sets are named and represented using capital letters. In the set theory, the elements that a set comprises can be any kind of thing – people, letters of the alphabet, numbers, shapes, variables, etc.
A set is a well-defined collection of distinct objects. Each object is said to be an element (or member) of the set.
We give below a few more examples of sets used particularly in mathematics, viz.
N : the set of all natural numbers
Z : the set of all integers
Q : the set of all rational numbers
R : the set of real numbers
Z+: the set of positive integers
Q+: the set of positive rational numbers, and
R+: the set of positive real numbers.
Elements of Sets
The objects present in a set are called elements of sets. The elements of sets are retained in curly frames separated by commas. To denote that an element is held in a set, the symbol ‘∈’ is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol ‘∉’. Here, 3 ∉ A.
Method of Represent Sets
Method of Represent Sets there are two methods:-
- Tabular Method (or Roster Method).
- Selector Method (or Rule Method or Set Builder Method).
Tabular Method or Roster Method
A set is denoted by a capital letter, i.e. A, B, X, Y, P, Q, etc. The general way of designing a set is writing all the elements (or members) within brackets ( )or { } or [ ].
The same set A can be taken as A = {blue, green, red, red, red}.
Selector Method (or Rule Method or Set Builder Method)
In this method, if all the elements of a set possess some common property, which distinguishes the same elements from other non-elements, then that property may be used to designate the set.
B = {x : x is an odd positive integer and 3 £ x £17}
Types of Sets
Types of sets are explained here:-
1. Finite Set
It is a set consisting of a finite number of elements. e.g. : A = {1, 2, 3, 4, 5}; B = { 2, 4, 6, ….., 50}; C = { x : x is number of student in a class}.
2. Infinite Set
A set having an infinite number of elements is called an Infinite set. e.g. : A = { 1, 2, 3, …..} B = { 2, 4, 6, ……} C = { x : x is a number of stars in the sky}.
3. Null or empty or Void Set
It is a set having no element in it, and is usually denoted by f (read as phi) or { }. For Example – The number of persons moving in air without any machine. A set of positive numbers less than zero.
A = { x : x is a perfect square of an integer 5 < x < 8}.
B = { x : x is a negative integer whose square is – 1}
Remember
(i) f ¹ {f}, as {f} is a set whose element is f.
(ii) f ¹ {0} is a set whose element is 0.
4. Equal set
Two sets A and B are said to be equal if all the elements of A belong to B and all the elements of B belong to A i.e. if A and B have the same elements.
For example:-
A = { 1, 2, 3, 4} : B = {3, 1, 2, 4}, or, A = {a, b, c} : B {a, a, a, c, c, b, b, b, b}.
[Note: The order of writing the elements or repetition of elements does not change the nature of the set]
Again let A = { x : x is a letter in the word STRAND} B = { x : x is a letter in the word STANDARD} C = { x : x is a letter in the word STANDING}
Here A = B, B ¹ C, A ¹ C
5. Equivalent Set
Two sets are equivalent if they have the same number of elements. It is not essential that the elements of the two sets should be the same.
For example:-
A = {1, 2, 3, 4} B = { b, a, l, 1}. In A, there are 4 elements, 1, 2, 3, 4, In B, there are 4 elements, b, a, I,1 (one-to-one correspondence), Hence, A º B (symbol º is used to denote equivalent set)
6. Sub-set
A set N is a subset of a set X if all the elements of N are contained in/members of the larger set X.
For Example:-
If, X = {3, 5, 6, 8, 9, 10, 11, 13} And, N = {5, 11, 13} Then, N is a subset of X. That is, N Í X (where Í means ‘is a subset of).
7. Proper Sub-set
If each and every element of a set A are the elements of B and there exists at least one element of B that does not belong to A, then set A is said to be a proper subset of B (or B is called a super-set of A). Symbolically, we may write, A Ì B (read as A is the proper sub-set of B)
And B Ì A means A is a super-set of B.
8. Power set
The family of all sub-set of a given set A is known as a power set and is denoted by P(A).
For example:-
(i) If A = {a}, then P(A) = {{a}, f,}
(ii) If A = {a, b}, then P(A) = {{a}, {b}, {a, b}, f.}
(iii) If A = {a, b, c}. P (A) = {{a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}, f.}
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 of sub-sets is 4 = 22 and when it is 3, the number of sub-sets is 8 = 23. So, if A has n elements, P(A) will have 2n sub-sets.
9. Universal Set :
In mathematical discussion, generally, we consider all the sets to be sub-sets of a fixed set, known as a Universal set or Universe, denoted by U. A Universal set may be finite or infinite.
For example:-
(i) A pack of cards may be taken as a universal set for a set of diamonds or spades.
(ii) A set of integers is a Universal set for the set of even or odd numbers.
10. Cardinal Number of a set
The cardinal number of a finite set A is the number of elements of set A. It is denoted by n{A). e.g. : If A = {1, m, n}, B = {1, 2, 3} then n(A) = n(B).
11. Universal Set
Usually, in a particular context, we have to deal with the elements and subsets of a
basic set that is relevant to that specific context. For Example, while studying the
system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”.
Types of Sets. Method of Represent Sets, Elements of Sets, Meaning of Sets.
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