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# Relations And Functions

In mathematics Relations and Functions play a very important role. Relation can be represented in the roster form and tabular form for example a relation ‘R’ on Set A = 1, 2, 3, 4,5 defined by R = ( a, b) : b = a + 2 can also be expressed as :-
a R b if and only if b = a + 2,
Let ‘A’ and ‘B’ the two Sets. Then a relation ‘R’ from set ‘A’ to set ‘B’ is a subset of A x B. Thus ‘R’ is a relation from A to B ↔ R is subset A x B. If ‘R’ is a relation from a non void set ‘A’ to a non void set ‘B’ and if (a, b) ε R, then we write a R b, which is read as 'a’ is related to ‘b’ by the relation ‘R'. If (a, b) ε R, then we write ‘a ! R b’ and we can say that ‘a’ is not related to ‘b’ by the relation ‘R’.
Domain:- Let ‘R’ be a relation from a set ‘A’ to set ‘B’. Then the set of all first components or coordinates of the ordered pairs belonging to ‘r’ is called the Domain of ‘R’.
Thus, domain of R = a: (a, b) ε R
Range:- Let ‘R’ be a relation from a set ‘A’ to set ‘B’. Then the set of all second components or coordinates of the ordered pairs belonging to ‘R’ is called The Range of ‘R’.
Thus, range of R = b: (a, b) ε R,
Types of relation are:-
-Void, Universal, And Identity Relation.
-Reflexive Relation.
-Symmetric Relation.
-Anti Symmetric Relation.
-Transitive Relation.
Function :- If f : A → B is a function, then ‘f’ associates all elements of set ‘A’ to elements in set ‘B’ such that an element of set ‘A’ is associated to a unique element of set ‘B’. Let ‘A’ and ‘B’ be two non empty sets, then a function ‘f’ from set ‘A’ to set ‘B’ is a rule which groups elements of set ‘A’ to elements of set ‘B’ such that :-
(i) All elements of set ‘A’ is grouped to elements in set ‘b’.
(ii) All elements of set ‘A’ is grouped to unique elements in set ‘b’.
Some standard real Functions that are widely used when we study Calculus are:
-Constant Function.
-Identity Function.
-Modules Function.
-Greatest Integer Function.
-Smallest Integer Function.
-Fractional Part function.
-Signum Function.
-Exponential Function.
-Logarithmic Function.
-Reciprocal Function.
-Square Root Function.
-Square Function.
-Cube Function.
-Cube Root Function.
Addition, product, difference, quotient, multiplication of a function by a Scalar, reciprocal of a function operation are performed on above functions.
Types of function are:-
-One One Function also called Injection.
-Many One Function.
-Onto Function also called Surjection.
-One One On to Function also called Bijection.
This was all about relations and functions.

## Functions

Algebra functions are one of the most important concepts of mathematics. A function can be considered as a rule which produces new elements out of some given elements. There are many terms such as 'map' to denote a function.
A relation ‘f’ from a Set ‘A’ to a set ‘B’ is said to be a function, if every element of set ‘A’ has one and only one image in set ‘B’.

## Relations

Relation Algebra is the algebra of Sets that deals with a Set of finite relations that is closed under some operators. These operators work upon one or more relations.
In mathematics a relation is a set of ordered pairs and the symbol for the set is denoted by “”. There are some examples of relations,
(0, 2), (55, 21), (3, 50),
(0, 2), (2, 5), (3, 9),
A relation is a set o...Read More

## Graphing Equations

A linear can be represented by linear equation by the use of graph and y – intercept and graph equations can be drawn as per the following steps
First need to locate the y – intercepts on the graph and then need to plot a Point.
From the created point above Slope is used to find a second point and then need to plot that point also.
Now the last step is to draw the ...Read More

## Tables and Relations

According to the relational model of database a table refers a convenient representation of a relation and its not necessary that the two should be equivalent like for example a SQL table can have duplicate rows but actually Relations does not allow duplication. Representation as a table implies a particular ordering to the rows and columns where as a relation is ex...Read More

## Patterns

There are many Algebra patterns defined in mathematics like solution of operations like addition, subtraction, multiplication and division.
Addition operation pattern: If any addition operation pattern is define like 2 * a + 6, here result of this addition pattern are depend upon value of a means when value of a changes, result is also changes –
For a = 1, result is 2 * a + 6...Read More

## Coordinate Plane

A coordinate plane is called the plane of two dimensions where we represent the point in the form of x- axis and the y- axis.  The coordinate planes, represent the numbers ‘a’ and ‘b’, which are in the specific order where ‘a’ is at the first place and b is at the second place of the Ordered Pair (a, b). In the coordinate plane we used to represent each Point in the pla...Read More