Question

Express the range of each function using roster notation

Let A = {1, 2, 3}.
f: A × A → Z×Z, where f(x,y) = (x,y+1).

Answer 1

To find the range of the function f(x, y) = (x, y+1), we add 1 to each element of the set A = {1, 2, 3} for the second component, which yields the set {2, 3, 4}. The range in roster notation is {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}.

Step-by-step explanation:

To express the range of the function f given by f(x,y) = (x,y+1), where the domain is A × A and A is the set {1, 2, 3}, we first determine the possible outputs of the function.

Since the first component of the output tuple is simply x, which comes directly from the set A, the possible values for the first component remain 1, 2, and 3. The second component is y+1, which depends on values from the set A. Adding 1 to each element of the set A, we get the possible values for the second component as 2, 3, and 4.

Now, combining each element of A with each of the possible values of the second component, the range of f in roster notation would be:

{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

This set represents all possible tuples that can be outputs of the function f, where f: A × A → Z×Z.