For f={(-4,4),(-2,4),(1,3),(3,5),(4,6)} and g={(-4,2),(-2,1),(0,2),(1,2),(2,2),(4,4)}

Find
a) f+g
b) f-g
c) g-f
d) fg

To find the operations between sets f and g, we need to understand the operations for each pair of corresponding elements in the two sets.

a) To find f + g:
We add the corresponding y-values of each pair of elements that share the same x-value. If an x-value in one set does not have a corresponding x-value in the other set, we ignore it.

f = {(-4,4), (-2,4), (1,3), (3,5), (4,6)}
g = {(-4,2), (-2,1), (0,2), (1,2), (2,2), (4,4)}

Looking for matching x-values:
(-4,4) + (-4,2) = (-4, 4+2) = (-4, 6)
(-2,4) + (-2,1) = (-2, 4+1) = (-2, 5)
(1,3) + (1,2) = (1, 3+2) = (1, 5)
(3,5) does not have a matching x-value in g.
(4,6) + (4,4) = (4, 6+4) = (4, 10)

Thus, f + g = {(-4, 6), (-2, 5), (1, 5), (4, 10)}.

b) To find f - g:
We subtract the corresponding y-values of each pair of elements that share the same x-value. If an x-value in one set does not have a corresponding x-value in the other set, we ignore it.

f = {(-4,4), (-2,4), (1,3), (3,5), (4,6)}
g = {(-4,2), (-2,1), (0,2), (1,2), (2,2), (4,4)}

Looking for matching x-values:
(-4,4) - (-4,2) = (-4, 4-2) = (-4, 2)
(-2,4) - (-2,1) = (-2, 4-1) = (-2, 3)
(1,3) does not have a matching x-value in g.
(3,5) does not have a matching x-value in g.
(4,6) - (4,4) = (4, 6-4) = (4, 2)

Thus, f - g = {(-4, 2), (-2, 3), (4, 2)}.

c) To find g - f:
We subtract the corresponding y-values of each pair of elements that share the same x-value. If an x-value in one set does not have a corresponding x-value in the other set, we ignore it.

f = {(-4,4), (-2,4), (1,3), (3,5), (4,6)}
g = {(-4,2), (-2,1), (0,2), (1,2), (2,2), (4,4)}

Looking for matching x-values:
(-4,2) - (-4,4) = (-4, 2-4) = (-4, -2)
(-2,1) - (-2,4) = (-2, 1-4) = (-2, -3)
(0,2) does not have a matching x-value in f.
(1,2) - (1,3) = (1, 2-3) = (1, -1)
(2,2) does not have a matching x-value in f.
(4,4) - (4,6) = (4, 4-6) = (4, -2)

Thus, g - f = {(-4, -2), (-2, -3), (1, -1), (4, -2)}.

d) To find fg (also known as the cartesian product):
We combine each pair of elements, taking one element from f and one element from g, resulting in all possible combinations.

f = {(-4,4), (-2,4), (1,3), (3,5), (4,6)}
g = {(-4,2), (-2,1), (0,2), (1,2), (2,2), (4,4)}

Taking one element from f and one element from g:
{(-4,4), (-4,2), (-2,4), (-2,1), (1,3), (1,2), (3,5), (4,6), (4,2), (4,4)}

Thus, fg = {(-4,4), (-4,2), (-2,4), (-2,1), (1,3), (1,2), (3,5), (4,6), (4,2), (4,4)}.