Let N={1,2.....n} .Define the Power set of N,P(N), and show that the map f:P(N)->P(N)
defined by taking A to belong to P(N) to N\A is a bijection.
Also prove using the above that
(n)=(n)
(k-1) (k)
Now the power set is defined by P(N)=2^n and a bijection is a one-to-one function that is both injective and surjective.But I don't understand the rest.I"ve been looking at it for the past 1 hour.
That is an interesting ideea...I never thought of using the complement.How can I use that to prove the last binomial relation from the problem?I know that
(n)=(n-1) + (n-1)
(k) (k-1) (k)
but is states to use part a)
For part a) my ideea for to use the fact that the domain and range have the same cardinality since we are using the power set of N but I don't know how to expres this in a mathematical way that will be inteligible.
I"m sorry maybe I'm just tired...but I"m more confused about both parts now that I was before.How can I use the complement in part a) to prove injection and surjection and what symmetry are you referring to?