1. For promotion to lieutenant, police officers in a city police department first have to pass a written aptitude test. This requirement was challenged in court by several black officers on the grounds that the test unfairly discriminated against blacks. In order to win the case, what did the black officers have to prove? What did the city have to prove, if anything, to be found innocent of discrimination? Solution They need to provide supporting documents showing that they have qualified with the written test, but was not qualified due to discrimination. Any discrimination should not be shown in the city to prove it. Also, should not support the accused..

1. 1. For sets A and B, find a necessary and sufficient condition for (A X B) intersection (B X A) =
empty set. Verify that this condition is necessary and sufficient.
Solution
1)
The condition is that sets A and B cannot contain any of the same elements. I'll try to show you
through some examples:
Ex1: A = {1,2} and B = {0,4}. Then A x B = {(1,0), (1,4), (2,0), (2,4)} and B x A = {(0,1), (4,1),
(0,2), (4,2)}. Then (A x B) (B x A) = (empty set). So it worked out because A and B didn't
contain any of the same elements.
Ex2: A = {0,1} and B = {0,3}. Then A x B = {(0,0), (0,3), (1,0), (1,3)} and B x A = {(0,0), (3,0),
(0,1), (3,1)}. Then (A x B) (B x A) = {(0,0)}, which is not the empty set so this doesn't work
because A and B contained at least one of the same elements.
But since we just looked at some examples like this doesn't mean that it's necessarily true in all
cases. So we need to prove it.
Proof: Let a A and b B. Two cases: (i) Suppose a = b, then A x B = {(a,b)} = {(a,a)} = {(b,b)}
and B x A = {(b,a)} = {(a,a)} = {(b,b)}. Then (A x B) (B x A) = {(a,a)} . (ii) Suppose a b, then
A x B = {(a,b)} and B x A = {(b,a)} and (A x B) (B x A) = .
So we can say from this that since (A x B) (B x A) = is necessary and sufficient if A and B are
disjoint (contain none of the same elements). Or in other terms (A x B) (B x A) = A and B are
disjoint.
2) For this you just need to give an example to validate that (AxB)c Ac x Bc.
You can do a lot of examples for this. One would be A = {0} and B = {1}. Then (A x B) =
{(0,1)} and (A x B)C = {All points except (0,1)}. Ac = {All numbers except 0} and Bc = {All
numbers except 1}. So Ac x bc = {All the points except (0,1)}.