We want to show that :
Part I: TPT:
Proof of Part I: let
Then, . By definition of symmetric difference,
Hence, ,
, OR
,
That is, .
Hence, , and
Hence, and
.
Hence, .
Hence, ,
, but
.
Therefore, , but
. —- Call this I.
Consider .
Therefore, .
Therefore, , and
Therefore, , and
.
Therefore, .
Hence, ,
, but
.
That is, ,
. Call this II.
From I and II, .
Part II: TPT: .
Quite simply, reversing the above steps we prove part II.
QED.
Cheers,
Nalin Pithwa