element. Definition The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. Definition The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R.
Note that range ⊂ codomain. Remarks (i) A relation may be represented algebraically either by the Roster method or by the Set-builder method . (ii) An arrow diagram is a visual representation of a relation. Example Let A = { , , , , , }.
Define a relation R from A to A by R = {( x , y ) : y = x + } (i) Depict this relation using an arrow diagram. (ii) Write down the domain, codomain and range of R. Solution (i) By the definition of the relation, R = {( , ), ( , ), ( , ), ( , ), ( , )} . Fig .
RELATIONS AND FUNCTIONS The corresponding arrow diagram is shown in Fig . . (ii) We can see that the domain ={ , , , , ,} Similarly, the range = { , , , , } and the codomain = { , , , , , }. Example The Fig .
shows a relation between the sets P and Q. Write this relation (i) in set-builder form, (ii) in roster form. What is its domain and range? Solution It is obvious that the relation R is “ x is the square of y”.
(i) In set-builder form, R = {( x , y ): x is the square of y, x ∈ P, y ∈ Q } (ii) In roster form, R = {( , ), ( , – ), ( , ), ( , – ), ( , ), ( , – )} The domain of this relation is { , , }. The range of this relation is {– , , – , , – , }. Note that the element is not related to any element in set P. The set Q