Relation and Functions

The topic relation and function deal with the problems of sets, so to understand Relation and Functions you should familiar with topics like sets, subsets.

Like the relation will tell you, about two following sets such as Input set and Output set.

whereas the function will let you know about, the relation which derives one output for each given input.

NOTE: Every function is a relation, but not all relations are functions. 

Let’s brush up on the basics:

TWO SETS:

• Input set
• Output set

Cartesian product of 2 sets is A*B(cross product)

where A is our Input set and B is our Output set.

• Above, element of A maps to element of B.
• So for our example: A ={2,3,4},  B ={1,5,6}
• Now our A*B ={(2,5),(3,5),(4,1),(3,6)}

Now for any subset of A*B is a Relation from A to B.

For eg. : {(2,5),(4,1)} is a relation.

SOME POINTS TO REMEMBER

1. In relation, one input can have multiple outputs.
2. In a relation R in the form (x,y), x is an element from the input set & y is from the output set.
3. We know this, Relation is called a function if an input has unique output. 
4. If a relation is defined on A*B, if the relation has more elements than the number of elements of A then, it cannot be a function. 

We will deal with A*A relation and types of relations.

Number of Relations  = number of subsets of A*A

Number of Elements in(A*A) = m*m =m²

Now a number of subsets = 2^k

(where  k = number of elements in that set)

therefore, a possible number of relations for set A  which having m elements = 2^m2.

Also, read Types of Relations.