For any integer n > 1, if Al, A2, A3, ... , An, and B are any sets, then
(A1 -B) (A2 -B) ... (An -B) = (Al A2 A3 … An) -B.
I was able to prove for all sets A, B, and C, (A -B)(C -B)= (AC) - B, and for the above I know that's it's true based on the just mentioned proof but I am having a hard time actually proving it. What steps would I take? I am not looking for an answer just a push in the right direction. Thanks.
I figured out I would have to use mathematical induction and I am going to be honest I am not the best at doing that.
oops missing the intersections
(A1-B)∩(A2-B)∩⋯∩(An-B)=(A1∩A2∩⋯∩A
I am not good at proof either because it involves words. I got a D once in my exam to do the proving. I knew the result, but I used wrong words to explain a line or two. My professor marked the answer 0 from that.
Anyway, you need to show it from the base knowledge when (A-B)intersect(C-B) = (AintersectC)-B. Then add one more with (D-B) and show that it is still the same way. From there, you could add n to it and it is still the same. Also, A represents A1, C represents A2, and so on.
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