Question

Let A = {1, 2, 3, 4, 5} and the functions f :A $\to$ A and g:A $\to$ A be defined by
f (1) = 3, f (2) = 5, f (3) = 3, f (4) = 1, f (5) = 2;
g (1) = 4, g (2) = 1, g (3) = 1, g (4) = 2, g (5) = 3.
Then

• A. fog = {(1,1), (2, 3), (3, 2), (4, 5)}
• B. fog = {(1,1), (2,3), (3,3), (4,5), (5,3)}
• C. gof = {(1, 1), (2, 3), (3, 3), (4, 4), (5, 5)}
• D. gof = {(2, 2), (2, 3), (3, 1), (4, 1), (5, 1)}

Answer 1

Answer: (B)

We have
(gof) (1) = g (f (1)) = g (3) = 1
(fog) (1) = f (g (1)) = f (4) = 1
(gof) (2) = g (f (2)) = g (5) = 3
(fog) (2) = f (g (2)) = f (1) = 3
(gof) (3) = g (f (3)) = g (3) = 1
(fog) (3) = f (g (3)) = f (1) = 3
(gof) (4) = g (f (4)) = g (1) = 4
(fog) (4) = f (g (4)) = f (2) = 5
(gof) (5) = g (f (5)) = g (2) = 1
(fog) (5) = f (g (5)) = f (3) = 3
$\therefore $ gof = {(1, 1), (2, 3), (3, 1), (4, 4), (5, 1)}
and fog = {(1, 1), (2, 3), (3, 3), (4, 5), (5,3).