## What are Relations?

Relations and Functions are an integral part of Mathematics, which helps to define the different concepts, along with different types of specific valued functions along with their graphs. A relation can be mathematically defined as the linking or connection between two different objects or quantities.

Functions, on the other hand, can simply be defined as a special kind of relations. With the help of proper examples, a clear distinction between the two can be established. You can practice this chapter and all its topics thoroughly with the help of the Relations and Functions Worksheet.

### What are the types of Relations?

The different types of relations are classified on the basis of their relation to other elements. Some of these different relations are:

- Empty Relation
- Universal Relation
- Reflexive Relation
- Transitive Relation
- Equivalence Relation

*Empty Relation**: *In a set ‘A’ if there is a relation ‘R’ which exists between any two elements, it can be defined as an empty relation.

*Universal Relation**: *A relation ‘R’ which exists in a set ‘A’ where each and every element of ‘A’ is related to one another, is termed as a Universal Relation.

*Equivalence Relation**: *An equivalence relation is said to exist when there is a relation ‘R’ for a set ‘A’ where R is said to be transitive, reflexive and symmetric. To perfectly understand an equivalence relation it is important to consider these types of relations.

### What is a Binary Operation?

A binary operation can be defined as the set of operations such as addition, subtraction, division and multiplication that are usually carried out to an arbitrary set called ‘X’. The operations that ensue, in order to obtain a result for the following pair of elements such a, b from X to another element of X is called as a binary operation.

### Important Questions:

- Is this a binary operation if in the set {5, 6, 7, 8, 9} * is defined by p * q = L.C.M. of p and q? Concur your answer.
- Is * commutative? Is * associative? if * be the binary operation on N defined by a * b = H.C.F. of a and b. Can you show the existence for this binary operation on N?
- From the set {5, 6, 7, … , n}, Find the number of all onto functions to itself
- Given a nonempty set Q, consider P(Q) which is the set of all subsets of Q
- Show that the function f : Q → Q given by f(p) = p
^{3}is injective.

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