Sets Relation and Function INTRODUCTION and Types of sets
Introduction:
The concept of a set was developed by German mathematician George Cantor (1845-1918) .
We often talk about group or collection of objects. Surely you must have used the words such as team, bouquet, bunch, flock, family for collection of different objects.It is very important to determine whether a given object belongs to a given collection or not.Consider the following collections:
i) Successful persons in your city.
ii) Happy people in your town
iii) Handsome boys.
iv) Days in a week.
v) First five natural numbers.
5.1.1 Set : Definition:
A collection of well-defined objects is called a set.The object in a set is called its element or member.We denote sets by capital letters A,B,C. etc.The elements of a set are represented by small letters a, b, c, x, y, z etc. If x is an element of a set A we write x∈A, and read as ‘x belongs to A’. Ifx is not an element of a set A, we write x∉A, and read as ‘x does not belong to A.’
e.g. zero is a whole number but not a natural number. ∴ 0∈W (Where W is the set of whole numbers) and 0∉N (Where N is the set of natural numbers)
5.1.2 Representation of a set:
There are two methods of representing a set.
1) Roster or Tabular method or List method
2) Set-Builder or Rule Method
3) Venn Diagram
1. Roster Method:
In the Roster method, we list all the elements of the set within braces {,} and separate the elements by commas.
Ex : State the sets using Roster method.
i) B is the set of all days in a week.
B = {Monday, Tuesday, Wednesday,Thursday, Friday, Saturday, Sunday}
Let’s Note:
1) If the elements are repeated, they are written only once.
2) While listing the elements of a set, the order in which the elements are listed is immaterial.
2. Set-Builder Method:
In the set builder method, we describe the elements of the set by specifying the property which determines the elements of the set uniquely. Sets Relation and Function INTRODUCTION and Types of sets
Ex : State the sets using set-Builder method.
i) Y = {Jan, Feb, Mar, Apr, ...., Dec}
Y = {x/x is a month of a year}
ii) B = {1, 4, 9, 16, 25, .....}
B = {x/x∈N and x is a square}
3) Venn Diagram:
The pictorial representation of a set is called Venn diagram. English Logician John Venn introduced such diagrams. We can use triangles, circles, rectangles or any closed figure to represent a set.
In a Venn diagram the elements of the sets are shown as points
A = {1,2,3} B = {a,b,c,d,e,f} C = {4,5,6}
Fig. 5.1
5.1.3 Number of elements of a set:
The number of distinct elements contained in a finite set A is denoted by n(A).
Thus, if A = {5, 2, 3, 4}, then n(A) = 4 n(A) is also called the cardinality of set A.
5.1.4 Types of Sets:
1) Empty Set:
A set containing no element is called an empty or a null set and is denoted by the symbol φ or { } or void set.
e.g. A = {x/x∈N, 1<x<2} = { }
Here n(A) = 0
2) Singleton set:
A Set containing only one element is called a singleton set.
e.g. Let A be a set of all integers which are neither positive nor negative. ∴A = {0} Here n (A) = 1
3) Finite set:
The empty set or set which contains finite number of objects is called a finite set. e.g set of letters in the word 'BEAUTIFUL'
K = {B, E,A ,U, T, I, F, L}, n(K) = 7 K is a finite set
4) Infinite set:
A set which is not finite, is called an infinite set. e.g. set of natural numbers, set of rational numbers.
Note :
1) An empty set is a finite set. 2) N, Z, set of all points on a circle, are infinite sets.
5) Subset:
A set A is said to be a subset of set B if every element of A is also an element of B and we write A ⊆ B.
Note: 1) φ is subset of every set. 2) A ⊆ A, Every set is subset of itself.
6) Superset:
If A ⊆ B, then B is called a superset of A and we write, B ⊇ A.
7) Proper Subset:
A nonempty set A is said to be a proper subset of the set B, if all elements of set A are in set B and at least one element of B is not in A.
i.e. If A ⊆ B and A ≠ B then A is called a proper subset of B and we write A ⊂ B.
e.g. Let A = {1, 3, 5} and B = {1, 3, 5, 7}. Then, evey element of A is an element of B but A ≠ B. ∴ A⊂B, i.e. A is a proper subset of B.
Remark:
If there exists even a single element in A which is not in B then A is not a subset of B and we wrtie, A ⊄ B.
8) Power Set:
The set of all subsets of a given set A is called the power set of A and is denoted by P(A), Thus, every element of power set A is a set.
e.g. consider the set A={a,b}, let us write all subsets of the set A. We know that φ is a subset of every set, so φ is a subset of A. Also {a}, {b}, {a,b} are also subsets of A. Thus, the set A has in all four subsets. viz. φ, {a}, {b}, {a,b} ∴ P(A) = {φ, {a}, {b}, {a,b}}
Sets Relation and Function INTRODUCTION and Types of sets
9) Equal sets:
Two sets are said to be equal if they contain the same elements i.e. if A⊆ B and B ⊆ A .e.g. Let X be the set of letters in the word 'ABBA' and Y be the set of letters in the word 'BABA'.
∴ X = {A, B}, Y = {B, A} Thus the sets X and Y are equal sets and we denote it by X = Y
10) Equivalent sets:
Two finite sets A and B are said to be equivalent if n (A) = n (B)
e.g. A = {d, o, m, e} B = {r, a, c, k} n (A) = n (B) = 4 ∴ A and B are equivalent sets.
11) Universal set:
If in a particular discussion all sets under consideration are subsets of a set, say U, then U is called the universal set for that discussion.
e.g. The set of natural numbers N, the set of integers Z are subsets of real numbers R. Thus, for this discussion R is a universal set. In general universal set is denoted by U or X.
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