Suppose A = {a, b, c}. Let f : A → A be the function f = {(a, c), (b, c), (c, c)}, and letg : A → A be the function g = {(a, a), (b, b), (c, a)}. Find g ◦ f and f ◦ g

To find g ◦ f, we need to perform the composition of the functions g and f.

Step 1: Start with the function g = {(a, a), (b, b), (c, a)} and the function f = {(a, c), (b, c), (c, c)}.

Step 2: Apply the function f to the elements of A.

• For a, f(a) = c.
• For b, f(b) = c.
• For c, f(c) = c.

So, f ◦ g = {(a, c), (b, c), (c, c)}.

Step 3: Apply the function g to the elements obtained from step 2.

• For (a, c), g(c) = a.
• For (b, c), g(c) = a.
• For (c, c), g(c) = a.

So, g ◦ f = {(a, a), (b, a), (c, a)}.

Therefore, g ◦ f = {(a, a), (b, a), (c, a)} and f ◦ g = {(a, c), (b, c), (c, c)}.