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.