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Range and Codomain of a Relation


If R is a relation, then the range of A is the set of all second components of the relation R. And codomain is the whole set of all second components. Note Range Codomain.

For instance, let A = {1, 2, 4, 5} and B = {1, 4, 8, 16, 25, 64, 125, 216}. Define R from A to B by x R y if x3 = y. For this relation the range is {1, 8, 64, 125}

Consider the sets

Consider the relation defined from A to B

For this relation the range is

Note that the range is the subset of the codomain.

 

​Example
Find the domain and range of each of the following relations:
  1. R = {(x, y)|y x2, x, y R}
  2. R = {(x, y)|x2 + y2 1, x, y R}
Solution

(i) The equation y = x2 represents a parabola that opens upwards and y x2 represent all the points inside and on the parabola y = x2. See the above figure. Now, the domain consists of all x-values that can be taken. Thus, domain = {x|x R}. The range is the set of all non-negative y-values that can be taken, so range = {y| y 0}.

(ii) The equation x2 + y2 = 1 represents a circle with centre at origin and radius equal to one unit. The relation, x2 + y2 1 represents the points inside and on the unit circle x2 + y2 = 1. See the above figure It is clear from the figure that domain consists of all x-values such that -1 x 1. Similarly, range consists of all y-values such that -1 y 1.
 Domain = {x| - 1 x 1} and range = {y| -1 y 1}.




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