Question

A compliment \B compliment = B-A​

Answer 1

Answer:

How can I prove that (A-B) = B'-A' in a set?

First, i’ll mention here, how do the following 2 Set Operations, differ from each other. ‘Complement' & ‘Difference'

For finding the ‘ Complement' of two sets, one set should be super set, & the other should be its sub set. Then we select the elements , which are present in super set but not in sub set. This forms the complement of the sub set.

But there is no such restriction while finding the ‘ Difference' of two sets. We may find the difference of any 2 sets X & Y, one may be subset or may not be subset. X-Y = the set of elements, which are present in X but not in Y.

Now, To Prove: ( A - B) =

A−B implies complement of B relative to A.

i.e, if x∈A−B

=>x∈A and x∉B

=>x∉A′ and x∈B′

=>x∈B′−A′

=>A−B⊆B′−A′—————(1)

Let x∈B′−A′

=>x∈B′ and x∉A′

=>x∉B and x∈A

=>x∈A−B

=>B′−A′⊆A−B—————−(2)

From (1) and (2) we have,

B′−A′=A−B