### Friday, April 17, 2009

## Infinities and Aparna's bijective map

Infinity, as the name suggests, is that which is not finite. The concept is a bit scary, not easily understandable. First time when I heard about it in mathematics, I was thrilled. Later when my programs started entering into infinite loops due to bugs, I started hating infinity. I even thought of leaving my programs to run for infinite time trying to understand what infinite time means. Seriously, I was so stupidly curious at that time. Soon after I left programming, I started loving infinity again. As I thought more and more about infinity and after reading the quote by Einstein, "Two things are infinite: Universe and Human Stupidity; I am not sure about the former", I defined a relation between stupidity, love and infinity. To understand that, you must understand the several kinds of infinities.

There are infinite types of infinities. Basically one needs to understand 2 kinds to understand infinite kinds. First consider a countable infinite set N = {1,2....infinity}. Then 2N = {2,4,....infinity}, which is same as removing alternate numbers from N. But since we can define a bijection(injective & surjective map) from N to 2N as f(x) = 2x for all x belonging to N, the cardinality (number of elements) of 2N is same as that of N. Therefore, the two sets are of same infinite order. A power set of a set is the set of all its subsets. Cardinality of the power set of a set with n elements is 2 power n. Obviously there can't be a bijection from a set to its power set, because the power set has more (means 2 power n) elements. Following this logic, power set of N mentioned earlier has cardinality 2 power infinity. Hence 2 power infinity is the immediate bigger infinity than our good old simple infinity. Now consider the power set of the power set, cardinality of which is 2 power 2 power infinity. Extending this infinite number of times, we find that there are infinite types of infinities.

Recently I have come up with a bijective map.

Let X be a set defined as X = {stupidity, love} and Y = {infinity, 2 power infinity}.

f:X->Y defined as ,

f(stupidity) = infinity, -----(1)

f(love) = 2 power infinity

= 2 power stupidity {from (1)

### Wednesday, April 01, 2009

## Can you "C"?

*Caution: I am egocentric and egotistic. No, a better word for me is "narcissist". This is yet another blog centered around myself.*

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