SETS

Introduction

Set Theory, branch of mathematics that deals with the properties of well-defined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.

Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of some concrete problems regarding certain types of infinite sets of real numbers. A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set.

A Set is a collection of objects, known as elements or members of the set.

An element ‘a’ belong to a set A can be written as ‘a ∈ A’,  ‘a ∉ A’ denotes that a is not an element of the set A.

Each element, separated by a comma, and then put curly brackets around the whole thing.

Example

Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

Set of odd numbers: {..., -3, -1, 1, 3, ...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

Positive multiples of 3 that are less than 10: {3, 6, 9}

Note :

1.      Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.

2.      The elements of a set are represented by small letters a, b, c, x, y, z, etc.

3.      When we say an element a is in a set A, we use the symbol ∈ to show it  ‘a ∉ A’.

4.      ‘a ∉ A’ denotes that a is not an element of the set A.

And if something is not in a set use ∉

Example

In a set of even number E , 2 ∈ E  but 3 ∉ E

SOME OF THE DIFFERENT NOTATIONS USED IN SETS
SL. NO	NOTATION	DISCRIPTION
1	∈	Belongs to
2	∉	Does not belongs to
3	: or |	Such that
4	∅	Null set or empty set
5	n(A)	Cardinal number of the set A
6	∪	Union of two sets
7	∩	Intersection of two sets
8	N	Set of natural numbers = {1, 2, 3, ……}
9	W	Set of whole numbers = {0, 1, 2, 3, ………}
10	I or Z	Set of integers = {………, -2, -1, 0, 1, 2, ………}
11	Z+	Set of all positive integers
12	Q	Set of all rational numbers
13	Q+	Set of all positive rational numbers
14	R	Set of all real numbers
15	R+	Set of all positive real numbers
16	C	Set of all complex numbers

Two methods are used to represent sets

1.  Statement form

2.  Roster forms

3.  Set Builder Form

1. Statement form

In this representation, the well-defined description of the elements of the set is given. Below are some Examples 1: The set of all even number less than 10.

Examples 2: The set of the number less than 10 and more than 1.

2. Roster forms

In this representation, elements are listed within the pair of brackets {} and are separated by commas.

Example 1:  Let N is the set of natural numbers less than 5.

Solution: N = { 1 , 2 , 3, 4 }.

Example 2:  The set of all vowels in the English alphabet.

Solution: V = { a , e , i , o , u }.

Note : In roster form, the order in which the elements are listed is immaterial, while writing the set in roster form an element is not generally repeated

3. Set Builder Form

In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

Example :

In the set {2,4,6,8} , all the elements possess a common property, namely, each of them is a even number less than 10. Denoting this set by N

N = {x:x is a even number less than 10}

We describe the element of the set by using a symbol x (any other symbol like the letters y, z, etc. could be used) which is followed by a colon ":" . After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces.

Cardinal number of a set

The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by n(A).

Example :  A {x : x ∈ N, x < 5}

A = {1, 2, 3, 4}

Therefore, n(A) = 4

Example :  B = set of letters in the word ALGEBRA

B = {A, L, G, E, B, R}

Therefore, n(B) = 6

Types of sets

The Empty set

Singleton Set

Finite Set

Infinite Set

Equal Sets

Equivalent Set

Universal Set

Subsets

Proper Subsets

Superset

Proper Superset

Power Set

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