Question

Let A = {1, 2, 3, ..., 14}. Define the relation R from A to A by R = {(x, y): 3x – y = 0, where $ x,y \in A $ }. Write down its domain, co-domain or range.

Answer 1

Hint: Here, use the equation given and find all values of x and y exist in set A and satisfy the given equation. All values of x satisfy the equation and belong to set A will be domain and all values of y satisfy the equation and belong to set A will be co-domain or range.

Complete step-by-step answer:
It is given that the relation R from A to A is given by R = {(x, y): 3x – y = 0, where x, y ∈ A}.
It means that R = {(x, y): 3x = y, where x, y ∈ A}
From equation 3x = y, we have following points
 $ \Rightarrow $ For x = 1, y = 3 × 1 = 3
 $ \Rightarrow $ For x = 2, y = 3 × 2 = 6
 $ \Rightarrow $ For x = 3, y = 3 × 3 = 9
 $ \Rightarrow $ For x = 4, y = 3 × 4 = 12
Hence, R = {(1, 3), (2, 6), (3, 9), (4, 12)}
We know that the domain of R is defined as the set of all first elements of the ordered pairs in the given relation.
Hence, the domain of R = {1, 2, 3, 4}
Determine the co-domain
We know that the entire set A is the co-domain of the relation R.
Therefore, the co-domain of R = A = {1, 2, 3,…,14}
As it is known that, the range of R is defined as the set of all second elements in the relation ordered pair.
Hence, the range of R is given by = {3, 6, 9, 12}.

Note: In these types of questions, write values of x and y as ordered pairs, then you can easily find the domain and range. Remember that we cannot take a value of x greater than 4 as it will yield values outside set A.