Math Is Fun Forum

cadjeff

Member

Registered: 2007-05-08

Posts: 26

Cardinality - DOMINATION!

Hi everybody,

I've been working my way through some cardinality questions (and constructions of number systems), the peano axiom and cauchy sequence questions have been ok but i'm stuck on this domination one now.....

i) Let A and B be sets such that A is dominated by B. Show that P(A) is dominated by P(B).

ii) Hence (or otherwise) show that if A is equivalent to B then P(A) is equivalent to P(B).

Would appreciate any help,

thanks!!

cadular

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Cardinality - DOMINATION!

P(X) is the set of all subsets of X.
A is dominated by B, so every element in A is an element of B.

But that means that every subset of A is also a subset of B.
Hence, every element of P(A) is an element of P(B), and so P(A) is dominated by P(B).

For the second part, remember that X and Y are equivalent iff X dominates Y and Y dominates X.

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Why did the vector cross the road?
It wanted to be normal.