Question

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = { ( x , y ): 3 x – y = 0, where x , y ∈ A}. Write down its domain, codomain and range.

Answer 1

It is given that A={ 1,2,3,⋯,14 } and the equation is,

3x−y=0 y=3x (1)

Substitute the value x=1 in equation (1).

y=3×1 y=3

Substitute the value x=2 in equation (1).

y=3×2 y=6

Substitute the value of x=3 in equation (1).

y=3×3 y=9

Substitute the value of x=4 in equation (1).

y=3×4 y=12

Substitute the value of x=5 in equation (1).

y=3×5 y=15

Substitute the value of x=6 in equation (1).

y=3×6 y=18

Substitute the value of x=7 in equation (1).

y=3×7 y=21

As it is given that the relation R from A to A is defined as R={ ( x,y ):3x−y=0, where x,y∈A } ; therefore, the relation only contains the values that belong to A, i.e.,

R={ ( 1,3 ),( 2,6 ),( 3,9 ),( 4,12 ) }

It is seen that the values after 12 is not considered because at x=5 , the value comes out to be 15 which does not belong to A.

The set of all first elements of the ordered pairs in a relation R is called domain.

Domain of R={ 1,2,3,4 }

The set of all second elements of the ordered pairs in a relation R is called range.

Range of R={ 3,6,9,12 }

The whole set A is the co-domain of the relation R.

Codomain of R={ 1,2,3,4....14 }

Thus, the domain is { 1,2,3,4 } , range is { 3,6,9,12 } and co-domain is { 1,2,3,⋯,14 } .