Answer:
(a) and (b) are not true in general. Refer to the explanations below for counterexamples.
It can be shown that (c) is indeed true.
Step-by-step explanation:
This explanation will use a lot of empty sets just to keep the counterexamples simple.
Note that can well be smaller than
. It should be alarming that the question is claiming
to be a subset of something that can be smaller than
. Here's a counterexample that dramatize this observation:
Consider:
.
(an empty set, same as
.)
(another empty set.)
The intersection of an empty set with another set should still be an empty set:
.
The union of two empty sets should also be an empty set:
.
Apparently, the one-element set isn't a subset of an empty set.
. Contradiction.
Consider the same counterexample
(another empty set.)
Left-hand side:
.
Right-hand side:
.
Apparently, the empty set on the left-hand side is not the same as the
on the right-hand side. Contradiction.
Part one: show that left-hand side is a subset of the right-hand side.
Let be a member of the set on the left-hand side.
.
and
(the right arrow here reads "implies".)
and
and
.
and
.
.
Note that (set on the left-hand side) implies that
(set on the right-hand side.)
Therefore:
.
Part two: show that the right-hand side is a subset of the left-hand side. This part is slightly more involved than the first part.
Let be a member of the set on the right-hand side.
.
and
.
Note that is equivalent to:
, OR
, OR
- both
AND
.
However, implies that
AND
.
The fact that means that the only possibility that
is
.
To reiterate: if , then the assumption that
would not be true any more. Therefore, the only possibility is that
.
Therefore, .
In other words, .
.
Combine these two parts to obtain: .