Relation

Let A and B be two sets. Then, A×B is called the Cartesian product of A and B. Any subset of A×B is called a relation from A to B. A relation from A to B is denoted by R:A→B or simply by R. For example,

Let A = {1,2} and B = {3,4}

Then, AB = {(1,3),(1,4),(2,3),(2,4)}

Let, R = {(1,3),(2,3)}

Here, R is a subset of A×B. So, R is a relation in A.

Domain and Range of the Relation

A relation from set A to set B is defined by a subset of Cartesian product A× B under the certain condition. The relation is generally denoted by R: A → B.

The first set A is called Domain of relation.

Second set B is called range of the relation.

Representation of relation

Relation may be expressed in various ways. Some ways are:

a. Listing or ordered pair form
R= {(Hari,Sujit)(Uma,Sharmila)(Sagarmatha,Dhaulagiri)}
R = {(1,1),(2,2),(3,3)}

b. Table form
Let, A = {1,2,3} and B = {1,4,3}.
Define R = {(x,y):y=x²}
Then, the relation R can be represented as:

c. Set builder form
Let A = {1,2,3} and B = {4,5,6} , Define R = {(x,y):x=y} i.e. R= {(1,1),(2,2),(3,3)}

d. Graphical form
Let A = {1,2,3} and B = {1,2,3}, R={(x,y):y=x²}

e. Arrow diagram form
Let A = {a, b} and B = {1, 2}

Inverse of a relation

If R = {(a,b): a∈A and b∈B} is a relation, then there exists another relation {(b, a) b∈B and a∈A} which is known as the inverse relation of R.

it is denoted by R^-1.

Mathematically, R^-1= [(b,a) : b∈ B and A ∈ a}

For example, If R = {( 1,a), (2,b), (3,c)}, then

R^-1= {(a,1), (b,2), (c,3)}