nothing much really happened today. got a lot of hw done and etc. no new anime. sigh…<\/p>\n
but here’s something for you to ponder upon. in mathematics, 2 sets of numbers are said to be the same size if u can match up one # from each set all the way to the end. do u still follow? cause the next step will jump out like you crazily. at first i was shocked too and tried to disprove it, but according to mathematics if there exist ONE one-to-one function between 2 sets of #s, then the domain and range are the same size. think about this D:{set of all even #s} and R:{set of all odd #s) and with the function f(d) = d+1 = r; then that’s a one-to-one function mapping evens to odds; therefore both sets are the same size. what threw me off was when liam said that the set of integers and the set of natural numbers was the same size. i was like no way! the set of integers is like 2x the size of the set of natural #s. my grounds was that the set of integers include the set of natural #s (i’m still standing for this pt, although after our TA\/GSI told us that liam was correct, i’m just making a fool outta myself). the thing is we can map the set of integers onto the set of naturals with a one-to-one function. think of this function where:<\/p>\n
f(d) = {
\n (d+1)\/2 if d is odd;
\n (-d)\/2 if d is even;
\n}<\/p>\n
this function maps every single integer you give it, onto the set of all natural #s. think about another equation. the set of all real #s between 0 and 1 exclusive and the set of all real #s greater than 1. they are the same size although the set of it’s obvious (or as liam says, you’ve been fooled) that the set of real #s greater than 1 is larger than the set of #s between 0 and 1 exclusive. lets try the function f(x) = 1\/x. Throw in any real # greater 1 and you’ll get a corresponding real # between 0 and 1. throw in any # between 0 and 1 and you’ll get a # greater than 1. no matter what # you try, this will always work.<\/p>\n
now i come to my hypothesis. since you can use one function to prove this true, why can’t i use one function to prove it’s not true. such as the set of positive reals and the set of all reals. since i was easily fooled to see the set of reals is 2x the size of the set of positive reals, i went ahead and found the function f(x) = x*x. Given any real #, i will always get back a positive real #, but the phenomenon is that there is exactly 2x the # of inputs than outputs meaning i can get 9 with +\/-3. meaning i’ve found an equation which has shown that the size of all reals is actually twice the size of positive reals. liam says i can’t do this because you can easily find a function to disprove anything, but those are just specific cases. the main thing is that if there exists at least one function that maps it from one-to-one, the it’s the same size. i sort of get what he’s trying to lead at, but there is that gap that i see increasing as u approach infinity. anyone wanna prove or disprove this? hehe.<\/p>\n
for those that was bored with my semi-lecture, check out this nothing much really happened today. got a lot of hw done and etc. no new anime. sigh… but here’s something for you to ponder upon. in mathematics, 2 sets of numbers are said to be the same size if u can match up one # from each set all the way to the end. do … <\/p>\nlink<\/a><\/del> link<\/a>. that girl is amazing and this link was donated by AznEECS. g’nite ^_^x<\/p>\n","protected":false},"excerpt":{"rendered":"