- Prove that if
and
, then
.
- Prove that if
and
, then
.
- If
and
then we claim
.
- If
and
, then
.
- What can you say if
and
?
- Proof. Let
be any element in
. Then,
since
. Further, there is some
such that
(since
means that
is a proper subset of
). Then, since
we have both
. Hence,
(since every element of
is in
, and there is at least one element of
not in
, so
).∎
- Proof. Let
be any element of
. Then,
since
. Further,
implies
since
. Thus,
.∎
- Proof. This was established in part (a) since we didn’t need
in that proof. (Since
a proper subset of
guaranteed there was some
in
that wasn’t in
, and since
is a subset, proper or otherwise, of
, this
is in
; hence,
is a proper subset of
.)∎
- Proof. Since
we know every element of
is in
. Hence, if
, then
.∎
- If
and
, then we cannot conclude that
. For example, let
and
. Then,
(since
is the set
and
contains this set). However,
, but
(since
contains the set which contains 1, but does not contain the element 1).