Relations and Functions

 In our daily life, we come across many relations such as father-son, brother-sister, teacher-student and many more.

In mathematics too, we come across relations such as
. A is a subset of B
. line l is parallel to line m
. number mis less than number n.

In all these, we notice that a relation involves pairs of objects in certain order. This chapter deals with the study of relations and functions in mathematics.

Ordered pairs and Cartesian products

     An ordered pair is a pair of entries in the specific order. The two entries are separated by comma and enclosed within brackets.

Some Important Theorem

     

     

Relations

     A relation R is a non-empty sub-set of a cartesian product.

     A relation is a set of ordered pairs, i.e., R Ì A x B where A and B are two non-empty sets.

Types of Relations

     Following are the types of relations:
Reflexive Relation, Symmetric Relation, Transitive Relation, Equivalence Relation.

Functions (or Mappings)

     Let A and B be two non-empty sets. A function f from A to B is an association of every element of A to an unique element in B.
We write this as f : A g B.

Composition of two functions or Product of two functions

     Let f:AgB and g:AgB be two functions. Thus the composition of two functions f and g denoted by gof or fog is the function from A into C defined by gof = {(a,b) for some c Î B, (a,c) Î f and (c,b) Î g}.

Binary Operations

     binary operation: Let S be any non-empty set. An operation * is called a binary operation on S if " a, b Î S a * b Î S

     Commutative law: Let * be a binary operation on the set S. * is said to be associative in S if " a, b Î S a * b =b * a

Summary

     A pair of objects, written in a specified order is called an ordered pair.

     For the ordered pair (a,b), a is called the first element and b, the second element.