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[SOLVED] countable and uncountable

dwsmith

Well-known member
Feb 1, 2012
1,673
If $A$ is a countable set and $B$ an uncountable set, prove that $B - A$ is similar to $B$.


Case 1: $|A| = n\in\mathbb{Z}^+$
Since $B$ is uncountable, $|B| = 2^{\aleph_0}$.
Then $|B - A| = 2^{\aleph_0} - n = 2^{\aleph_0}$.
Therefore, $B - A$ is equinumerous to $B$, and hence $B - A$ is similar to $B$.


Case 2: $|A| = \aleph_0$
Again, we have $|B - A| = 2^{\aleph_0} - \aleph_0 = 2^{\aleph_0}$
Therefore, $B - A$ is equinumerous to $B$, and hence $B - A$ is similar to $B$.

Does this work?
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
What does "similar" mean?

Since $B$ is uncountable, $|B| = 2^{\aleph_0}$.
This is wrong. Also, even for finite sets, |B - A| is not necessarily |B| - |A|.
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
If $A$ is a countable set and $B$ an uncountable set, prove that $B - A$ is similar to $B$.
There is completely trivial proof if A is a subset of B.
You know that the union of two countable sets is countable.
You also know that $B=A\cup(B-A)$. What if $B-A$ were countable?
 
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