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Response Details:
To prove (A - C)  (B - C) = (A B) - C
We must prove (A - C) (B - C)  (A B) - C and (A B) - C (A - C) (B - C).
(A - C) (B - C) (A B) - C:
Assume (A - C) (B - C), then by definition of difference of sets x  A and x  C.
Also by difference of sets, x B and x C.
By definition of intersection, x A and x C and x B and x C.
In particular x A and x B and x C.
By definition of intersection, x A B.
And since x C then x (A B) - C.
As was to be shown.
(A B) - C (A - C) (B - C):
Assume (A B) - C (A - C) (B - C) to be true.
So x (A B) and x C by definition of difference of sets.
By definition of intersection, x A and x B.
So x A and x C, by difference of sets x (A - C).
Also x B and x C so by difference of sets x (B - C).
And since x (A - C) and x (B - C).
The by definition of intersection x (A - C) (B - C).
As was to be shown.
You can add in your own take on the explanation of steps, but this is a basic outline of how I would go about it.
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