Any subset of a Cartesian product A×B  in which the first element and second element of ordered pairs have special relation to each other is known as “Relation”.

A relation from one  set (A)  to another set (B) is denoted by:

“xRy” or simply “R”

Where (x,y)∈R

For example:

If set A={Me , My father , My son}

And set B={My spouse , My mother , My daughter}

Then one of the “Relation” from set A to set B can be:

R=[{My spouse , Me} , {My mother , My father}]

In above Relation , the relation between the first and second element of ordered pairs is that “first element is wife of Second element”. Like “My spouse” Is wife of “Me”

And

If set A={2,3,4}

And set B={4,5,6}

Then one of the “Relation” from set A to set B can be :

R=[{2,4} , {2,6} ,{3,6} ,{4,4}]

In above Relation the relation between first and second element of ordered pairs is that “First element is a factor of second element”. Like 4 is a factor of 4.

Domain and Range of a Relation:

“Domain” of a Relation(R) is the set of all the first elements of ordered pairs of the Relation(R)

and

“Range” of a Relation(R) is the set of all the second elements of ordered pairs of The Relation(R).

For Example:

If a Relation R=[{1,2} ,{2,3} ,{3,4}]

Domain of Relation R ={1,2,3}

And Range of Relation R={2,3,4}

Diagrammatically we can denote the relations from one set(A) to another (B) as following:

,

etc.

Inverse Relations:

A relation obtained by interchanging first and second elements in the ordered pairs of given Relation is known as the inverse Relation of given Relation.

If a Relation “R” is given then the inverse of the relation “R” is denoted by the symbol:

R^-1

For example:

If Relation R=[{1,2} , {3,4} , {5,6}}

Then the  inverse of Relation R =R^-1=[{2,1} , {4,3} , {6,5}]

The Cartesian Product of two sets A and B is the set of all Ordered Pairs (a,b)  where the first element of order pairs “a” belongs to first set “A” and second element of ordered pairs “b” belongs or second set “B”.

Or a∈A and b∈B

Note: Cartesian product of set A and B is not equal to Cartesian product   of set B and A.

Denotation of Cartesian product:

Cartesian product of sets “A” and “B” is denoted by :

A×B

And Cartesian product of sets “B” and “A” is denoted by:

B×A

For example:

If set A={1,2} and set B={4,5}

Then,

A×B=[ {1,4} , {1,5} , {2,4} , {2,5} ]

And

B×A=[ {4,1} , {4,2} , {5,1} , {5,2} ]

Note: If “m” is the number of elements in set “A” and “n” is number of elements in set “B” then the numbers of elements of A×B and B×A is m×n

For example:

If set A have 2 elements and Set B have 3 elements the the number of elements that A×B and B×A have is 3×2=6.

What is a pair?

A pair is a set  which have only two elements. Like {a,b} , {me,my sister}. A pair can never have more than or less than two elements ; it always have exactly two elements.

What is ordered pair?

A pair or set of two elements , in which order of its elements are predefined or the first element is always in first and second is always in second is known as an ordered pair.

An ordered pair having “a is its first element and “b” as its second element is denoted by:

{a,b}

Equal ordered  pairs:

As , order of elements of an ordered pair is predefined ,

Two ordered pairs {a,b} and {c,d} are equal if and only if “a”=”c” and “b”=”d”

If “a” and “b” are not equal to “c” and “d” respectively the two ordered pairs cannot be equal.

Note:

> A pair {a,b} is equal to another pair {b,a}

For example pairs {1,2}={2,1}

>An  ordered pair {a,b} is never equal to another ordered pair {b,a} unless a=b.

For example: order pairs {1,2} ≠ {2,1} but {1,1}={1,1}

So ,

What is Number?

Number is one of the basic concept of mathematics. A Number basically representation of quantity of any object. Number system started from the time of ancient civilization , Primitive man used to compare one quantity with another when there was no number ; Like they used to lay aside each pebble for a sheep they rear , in order to count sheep.  latter with advancement in language they gave the number a name and a symbol like for first quantity the name is “One”  and symbol is “1″ and for fifth quantity the name is “Five” and symbol is “5″.

Each civilization had its own  name and symbols for numbers , among them the most popular one now is “Hindu Arabic” numbering system which has 10 base (or this number system has 10 basic symbols for numbers).

The ten symbols used in “Hindu Arabic” numbering system for numbers are:

0, 1 , 2 , 3 , 4, 5 , 6 , 7 , 8 , 9

Note:

> In 10 base numbering system the higher number are represented by the combination of two or more number symbols to form a single number symbol liker to represent Ten  we combine “1″ and “0″ to form “10″ which means Ten.

> The set of numbers 1,2,3,4…. where each number has a successor is known as “Natural Numbers”

> Zero “0″ symbolizes no quantity.

In the course of time humans discovered other complex types of numbers like Negative  Numbers ; which developed after the concept of negative quantity or debt , Rational (Fractional) Numbers , Irrational (Non-fractional) Numbers.

So now,

What is Real Number?

The collection of sets of following types of numbers is known as real number.

a.  Natural Numbers.

Example: 0,1,2…….10,11,12……..100,101,102………

b. Integers.

Example:

Negative:………….-3,-2,-1

Zero: 0

Positive: 1,2,3………………

c. Rational Numbers.

Example: 0/1 , -1/2 , 3/4 , 12/-4………………….

d.Irrational Numbers.

Example: √2 , √3 , 3√3 , Π

We can also show the above relation between other types of numbers and Real Number diagrammatically by the following diagram:

Union:

A set that contains all the elements contained by first set (A) and second set (B) is known as union of the two sets (A and B).

We denote union of two sets (A and B) by symbol A ∪ B.

For example: if A={1,2,3} and B={3,4,5} Then,
A ∪ B={1,2,3,4,5}

Intersection:

A set whose elements are the common elements of two sets (A and B) is known as the intersection of the sets(A and B).  The intersection of two sets (A and B) is denoted by the symbol A ∩ B.

For example: If A={1,2,3} and B={2,3,4} Then A∩B={2,3}

Complement:

A set whose elements are all the elements of universal set except a set (A)is known as the complement of the set (A). The complement of a set (A) is denoted by symbol  and read as “A complement”

For example: If A={1,2,3} , B={3,4,5} and C={4,5,6,7} Then , Â={4,5,6,7}

Difference:

The difference of set A and B is the set formed by a set with all elements of set A that does not belongs to set B. We denote the difference of set A and B by A-B and difference if set B and A by B-A.

For example: If A={1,2,3,4} and B={3,4,5,6} then,

A-B={1,2} , B-A={5,6} , A-A= φ and B-B= φ

Above set operations are shown below as graphical representation in Venn diagram.

Union:

In the following figures A∪B is shown as shaded region:

Intersection:

In the following figures A∩B is shown as shaded region , in second figure no region is shaded because in the figure A∩B=Φ

Complement:

In the folowing figure  is shown by shaded region:

Difference:

In the first figure below A-B is shown as shaded region and in second figure A-A is shown and no region as shaded as A-A is Φ

For example, the order three diagrams consist of three mutually intersecting circles with eight regions. The region A, B and C consists of members which are only in one set not in others. The three regions consist of members which are in two sets but not in three. consists of member which are in all three sets and no region occupied represents .

Some of examples of venn diagram showing relation between sets are given below:-

The Venn diagram above represents B?A or set B is subset of set A.

The Venn diagram above represents A=B or set A is equal to set B.

The Venn diagram above represents that set A and set B are disjoint set.

The Venn diagram above represents that set A and set b are intersecting.

The Venn diagram above represents following things:
a> Set A and set B are intersecting.
b> Set B and set C are intersecting.
c> Set C and set A are intersecting.
d> Set A , B and C are also intersecting.

Subset:

If one set (A) contains  all the elements that another set (B) contains then the second set (B) is called to be the subset of first set (A) , or set B contains set A.

In symbol we write

A ⊂ B  (A is contained in B) , B ⊃ A (B contains A)

Both symbols above means that set A is a subset of set B.

A set may have two or more subsets.

For example: If set A={1,2,3,4} Then {1} , {2,3} , {4,1} etc. are the subsets of set A.

Note:

*The number of elements a set contains is known as its cardinal number.

*The number of possible subset a set can have is given by the formula 2^s , Where “s” is the cardinal  number  of set A.

*If a set contains all other set that are currently being discussed then the set is called universal set.

Equal set:

Two sets are said to be equal if every element contained by first set is contained by second set and also every elements contained by second set is contained by first set.

equal sets are  sub set of each other.

For example:

If set A={a,b,c,d} and set B={d,a,b,c} Then set A and set B are equal set.

Proper Subset:

If A ⊂ B and A ≠ B(A is not equal to B) then set A is said to be a proper subset of set B. In other words , A set is said to proper subset of another set if every elements contains by the set in contained by another also but the another one also contains some elements not contained in the first set.

For Example:  If A={1,2,3} and B={1,2,3,4} then set A is a proper subset of set B.

Power set:

A set of all subsets of any set is known as power set. It is denoted by “2^s”.

For example:  If S={a,b} then all possible subsets of set S are :  ø , {a} , {b} ,{a,b}

So , 2^s of set S is [ø , {a} , {b} ,{a,b}]

As told on the note above the cardinal number of  power set is given by formula 2^s where “s” is the cardinal number of any set.

Disjoint sets:

Two sets are said to be Disjoint if they dont have any common element.

For example: The set of boys and the set of girls is disjoint , If set A={1,2} and set B={3,4} then set A and B are disjoint.

Intersecting sets:

Two sets are said to be intersecting if some of elements they have are common in both.

For example: If set A={1,2,3} and set B={3,4,5} then set A and set b are  intersecting sets.

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The concept of modern mathematics is started with set. Set appears in all branches of mathematics. The main developer of set theory is George Cantor (1845-1915 AD).

George Cantor

The word set is synonym with “Collection” , “Class” or “Aggregate”. Basically set is a collection or organization of similar objects (an object may be material or conceptual). and the objects by which a set is made of  are called elements or member of the set. Some example of set are:
1> The countries of Europe.

2>The solar system.

3>The peoples living in my house.

4>Vowels of the English alphabets. etc.

Notation:

Sets are usually denoted by capital letters and the elements of set are denoted by small letters. For example : Set A={a,b,c,d} The symbol    denotes set membership and symbol   denotes non membership. For example in set A above, a   A but e  A , Which means the element “a” belongs to set “A” but the “e” doesn’t.

Specification:

A set can be denoted or membership of a set may be indicated in several ways. Two common ways among them are:

a> Listing or Tebulation. In this method elements of set are listed , sperated by commas and enclosed inside a bracket. For example: V={a,e,i,o,u} My family={me , my son , my spouse}

b>Description or Rule. In this method a set is specified by enclosing in brackets a descriptive phrase or a rule. For example: V={the vowels of the English alphabet.} N={x:x is a natural number} A={x:x^2-3x=0} In last two examples the respective set is a group of element x , where each value of element x is defined by the respective rule.

Finite and Infinite set:

If a set have a finite numbers of elements then a set is called finite set else the set is called infinite set. For example: D={0,2,4,6,8} A={a,e,i,o,u} The above two sets are finite set while the following sets are infinite one. The set of stars in Universe. D={set of natural numbers}

Null Set:

A set which has no element is called Null set. A Null set is also called Empty set or Void set. It is denoted by symbol ø  For example: A={x:x is a man who gave birth to a child) N={x:xis a natural number , x>0 and x<1 }

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