Question

If, A⊂B, Prove that B'⊂A' ?

If, A`sub` B,
Prove that B'`sub`A'

Answer 1

Please see below.

Explanation:

`A` is a set containing certain elements. How do we define `A'`?

Let us consider a universal set `U`, which is set of all the possible elements. Now as `A` contains some elements, `A'` is its complementary set which contains all the elements, other than those contained in `A`.

In other words `A subU`, `A'subU`, but `AuuA'=U` and `AnnA'=phi`. In other words `A` and `A'` together contain all the elements in the universal set `U`, but there is no element common between `A` and `A'`. In venn diagram we describe it as under:

Now as `AsubB`, let `B` and `B'` be as shown below.

Further as `AsubB`, we have `AnnB=A`

Comparing `A'` and `B'`, it is apparent that `A'` contains all the elements `B'` and additionally those as well which are contained in `B` but not `A`.

Hence `B'subA'`