show that (A U B) - (A n C) = B WHERE C = ~(B) IF A , B , C are finite sets and?

Question

we have to proove that in sets and relations prooving that using diagramsis very easy

Answer 1

(A U B) - (A ∩ C)

= (A U B) - (A ∩ ~B)

= [A + B - (A ∩ B)] - (A ∩ ~B)

= A + B - [(A ∩ B) + (A ∩ ~B)]

= A + B - A

= B

Answer 2

That's very easy to show with a diagram, but that's very hard to do with a text interface. Get pencil a paper and follow me.

Draw two circles which partially overlap. One circle represents A, the other B. The bit in the middle is AnB. The bit on the left is A-B, on the right is B-A.

Now An(~B) = A-B, the bit on the left. AuB is all three bits together. So (AuB) - (A-B) is all three bits take away the bit on the left.

What you have left is two parts that together make up B.