1. Consider three sets A, B, and C each with three elements, and define a function 'f' from set A to B and a function 'g' from set B to C with the mapping of the elements you specify.

• After defining 'f' and 'g', discuss the composition of functions fog and gof on the sets A, B, and C that you have defined. Can both fog and gof be defined for the sets you have considered?

Are they equal?

This is my answer: Is it correct?

Set A: {1, 2, 3}

Set B: {4, 5, 6}

Set C: {7, 8, 9}

Function f: A B

f(1) = 4

f(2) = 5

f(3) = 6

Function g: B C

g(4) = 7

g(5) = 8

g(6) = 9

Now, let's discuss the composition of functions fog and gof:

Composition fog: A C

(fog)(1) = g(f(1)) = g(4) = 7

(fog)(2) = g(f(2)) = g(5) = 8

(fog)(3) = g(f(3)) = g(6) = 9

Composition gof: A C

(gof)(1) = f(g(1)) = f(7) (Note: 7 is not an element of set B, so gof is not defined for this input)

(gof)(2) = f(g(2)) = f(8) (Note: 8 is not an element of set B, so gof is not defined for this input)

(gof)(3) = f(g(3)) = f(9) (Note: 9 is not an element of set B, so gof is not defined for this input)

Based on the above calculations, we can see that the composition fog is defined for all elements in set A, while the composition gof is not defined for some elements in set A. Therefore, fog is a valid composition, but gof is not defined.

As for the equality of fog and gof, since gof is not defined, we cannot compare them for equality.

In conclusion, fog can be defined for the sets A, B, and C considered, while gof is not defined. The two compositions are not equal.