freaking hot weather!!! TIAN YU!!! give me back my rain!! haha dun just keep it for urself in France!! haha ok i'm on the verge of going nuts... haha anyway today i think yang yue took us for set theory... and i must say he's damn good!! haha anyway got a very nice proof which i shall write here...

Cantor's Theorem: The cardinality of the power set of any set X is strictly greater than the cardinality of X.

Proof. First we show that card(X) less than or equals to card(P(X)). To do this, we need only prove that there exists an injection from X onto P(X). Let f:X->P(X) where f(x) = {x}. Clearly f is well defined and injective. Therefore card(X) less than or equals to card(P(X)). In order to show that card(X) cannot be equal to card(P(X)), we show that there cannot exist a surjection bewteen these 2 sets. We suppose, for the sake of contradiction, that g is one such function. To proceed, we need to show that there is an element in P(X) that does not lie in the image set of g on X (ie. we will show that there is a subset of X that does not lie in the range of g. We now define a set A which is a subset of X as follows: A={x ε X : x NOT ε g(x)}. Now, since g is surjective, A must lie in the range of g (ie there exisit an element a ε X such that f(a)= A. However, a ε A=f(a) => a ε f(a) but a ε f(a) => a NOT ε f(a) which is a contradiction. Similiarly if a NOT ε A=f(a) => a NOT ε f(a) but a NOT ε f(a) => a ε f(a) which is also a contradiction. Therefore for any set X, card(X) is strictly less than card(P(X)).

phew... i din think i would be able to provide the proof... haha finally understand it... hmm this seems like a good idea... haha writing down proofs that took me a while to understand... haha really drills it in... haha this one in particular is sooooo elegant!! haha... hmmm anywayz i think more proofs on analytic number theory shld be coming soon... haha really confused with that topic... hee... anyway... think i had enuff for tonight... frends coming up... haha... okie... time to rest... hee hee ciao!

overheated!!

had breakfast with gerri at ghim moh today...