Question

given a set A, is A = ∪P(A)?
Note: ∪P(A) is the union of the power set of A

Answer 1

Yeah.

Answer 2

You mean the union of the subsets of \(A\)?

Answer 3

Yes, that's clearly going to be \(A\).

Answer 4

oops yeah the union of the subsets of A

Answer 5

ok but how would you prove it though?

Answer 6

You can do the full proof, but A is just a union of its subsets. Quite intuitive.

Answer 7

Try something a bit more difficult. Like show that the cardinality of a set and its power set differ.

Answer 8

Let a be in \(A\), then \(\{a\}\subset A\) so \(\ a \in \large\cup_{\small B\in P(A)}B\)
Let \(a\in \cup_{\small B\in P(A)}B\) then \(a\in B\) for some \(B\subset A\) so \(a\in A\)

Answer 9

Try this one, suppose \(A\) is countably infinite, show that \(P(A)\) is uncountably infinite.

Answer 10

so \(\cup_{\small B\in P(A)}B\subset A\) and \(A \subset \cup_{\small B\in P(A)}B\) so \(A=\cup_{\small B\in P(A)}B\)

Answer 11

Hint: Use diagonalization

Answer 12

if you just need problems to solve, go solve mine:)

Answer 13

I'd rather do a simple proof. :D

thanks guys