Prove the following.
-
.
-
.
-
if and only if
.
- Proof. To show two sets are equal, we want to show that each is a subset of the other (i.e., we want to show that
and
).
First,since the only element of
is
, and we have
.
Second, the only element ofis
, and we have
. Hence,
. Thus,
. ∎
- Proof. Again, we want to show
and
.
First,since the elements of
are
and
, and
.
Second, the elements ofare
and
. Since
we have every element of
is in
; thus,
.
Therefore,. ∎
- Proof.
Assume
. Since
, we must have
; hence, every element of
must be contained in
. This means that both
and
are in
. Since
is the only element of
, we must have
.
Conversely, assume
. Then
and from part (a) we know
; hence
.∎