SETS, RELATIONS & FUNCTIONS

SETS, RELATIONS & FUNCTIONS

Sets are a fundamental part of mathematics and it’s knowledge is very important in present. It defines the concepts of relation and functions. These functions are also very essential for present day maths.

SETS

Sets are denoted by capital letters. They are similar to matrices whose elements are denoted by small letters. But in sets an element comes only at once. Sets are matrices which give a sense of belonging. This sense of belongingness is represented by ∈  .We can write anything in a set like rivers of India, Wonders of the World .etc. 

For an example let there be a set of positive single digit. Let the set be denoted as A

Therefore, 8 ∈ A

 Also we can write, 17 ∈ A

Set can be represented by two types 

ROSTER

In roster form we write elements in curly brackets and separate them with comma. 

If there be set of positive integers less than 10. Therefore the set will be {1,2,3,4,5,6,7,8,9} 

SET- BUILDER FORM

In set-builder form we take a variable x and then gives its condition. For eg: we can write the above example as

{x: x is positive integer less than 7}

There are some standard symbols for specific sets like

N : set of all natural numbers

Z : set of all integers numbers

R : set of all real numbers

Z⁺/Z^- : set of all positive/negative integers

Q⁺/Q^- : set of all positive/negative rational numbers

R⁺/R^- : set of all positive/negative real numbers

TYPES OF SETS 

There are various kind of sets. 

EMPTY SET

Set having no element is called an empty set

 For eg: {x: x < 0, x is a whole number }

Whole number is 0 and more than 0. Thus it is a empty set

FINITE AND INFINTE SETS

Sets which have limited elements are called finite set 

For ex: {x: 0 < x < 15}

Sets which have infinite number of sets are called infinite set 

For ex: {x: x ∈ R}

EQUAL SETS

Sets which have same elements are called equal sets. The elements can be in any order

For ex: {2, 5, 6, 7} and {5, 7, 6, 2} are equal sets

             {x: 0<x<11} and {x: x is an natural number less 11}

NOTE : in this articles I can use the following sign

              ≤ - more than or equals to 

              ≥ - less than or equals to

SINGELTON SETS

Sets having only single element are called singleton sets. 

For ex: {x: 1<x<3, x ∈ Z} is an singleton set.

SUBSET

Subset means when every element of a set A is in another set B. In this case we write … 

A ⊂ B

If a ∈ A and A ⊂ B then a ∈ B

if A is not a subset of B then we write

A ⊄ B

For ex: A = {1, 3, 5} , B = {2, 3, 1, 6, 5, 7} then A ⊂ B

If A ⊂ B and B ⊂ A then the two sets are equal

Remember A is a subset of B is not vice versa, which means that if A is a subset of B then it is not necessary that B is a subset of A.

Also, 

• Every set is a subset of itself
• Empty set is a subset of every set
• The maximum number of subsets of a set having n elements is 2ⁿ

INTERVAL AS SUBSETS OF SETS OF REAL NUMBERS   

Let a, b ∈ R. Let a set is in the form

{y: a<y<b}

Then this is known as an open set which means that ‘a’ and ‘b’ are not included in the set. Such sets can be written as (a, b) with curly brackets.

If ‘a’ and ‘b’ are included in the set then we write [a, b] with square brackets

There can be set known as the semi-open set. In this set it is open at one end and closed at the other. Like … [a, b)…closing at ‘a’ and opening at ‘b’ or …(a, b]… opening at ‘a’ and closing at ‘b’

This can be represented in a number line with the help of filled and unfilled dots. Filled dots for closed and unfilled for opened 

POWER SET OF A SET

Power set is a set containing all the subsets of a set A. It is denoted by P(A). In P(A) all elements are sets. If A has n elements then P(A) has 2ⁿ elements

If A = {a, b} then P(A) = {φ, {a}, {b}, {a, b}}

UNIVERSAL SET

Out of given number of set if there is a set such that every other set is a subset of it then it is called a universal set

If there is are sets A, B, C, D

Such that A ⊂ D, B ⊂ D, C ⊂ D then D is the universal set. It is represented by ‘U’

OPERATION OF SETS

UNION OF SETS

Union sets means that a set which contains all the elements of A + all the elements of B. If A and B have some common elements then we do not repeat them.

It can be represented by

A ∪ B {x: x ∈ A or x ∈ B}

The colored region is the desired set. 

INTERSECTION OF SET

Intersection of set is a set which have only the common elements of the two sets

It can be represented by , A ∩ B

the yellow region is the intersection of A and B

COMPLEMENT OF A SET

Complement of a set is the difference of the set and its union set

It is denoted by A^’

     A + A^’ = U