Domain and image

Definition: Let R be a relation of A to B. The set D of all the first elements of the ordered pairs belonging to R is called the domain of R.

x

The set Im of all the second elements of the ordered pairs belonging to R is called the image of R.

y

It follows from the definition that DCA, and ImCB.

Examples:

  1. If A = {0,2,3,4} and B = {1,2,3,4,5,6} what is the domain and the image of the relation R = {(x,y) E A X B y is a multiple of x}? Using the arrow scheme, it is easy to see that D is the set of elements of A from which arrows depart and that Im is the set of elements of B to which arrows arrive, meaning:

R = {(2.2), (2,4), (2,6, (3,3), (3,6) (4,4),)} D = {2,3,4} Im = {2,3,4,6}

  1. If A = {x} and B = {y}, what is the domain and range of the relation of R = {(x,y)}? Using the Cartesian representation we have D = {x} and Im = {y}