MHB Is $B - A$ always similar to $B$ if $A$ is countable and $B$ is uncountable?

  • Thread starter Thread starter Dustinsfl
  • Start date Start date
Click For Summary
If $A$ is a countable set and $B$ is an uncountable set, it is proven that $B - A$ is similar to $B$. In the first case, when $|A|$ is a finite integer, the cardinality of $B - A$ remains $2^{\aleph_0}$, confirming its equinumerous nature to $B$. In the second case, when $|A|$ is countably infinite, the same conclusion holds as $|B - A|$ also equals $2^{\aleph_0}$. The discussion highlights that removing a countable subset from an uncountable set does not change its cardinality. Thus, $B - A$ is indeed similar to $B$.
Dustinsfl
Messages
2,217
Reaction score
5
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?
 
Physics news on Phys.org
What does "similar" mean?

dwsmith said:
Since $B$ is uncountable, $|B| = 2^{\aleph_0}$.
This is wrong. Also, even for finite sets, |B - A| is not necessarily |B| - |A|.
 
dwsmith said:
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?
 
Last edited:
Hello, I'm joining this forum to ask two questions which have nagged me for some time. They both are presumed obvious, yet don't make sense to me. Nobody will explain their positions, which is...uh...aka science. I also have a thread for the other question. But this one involves probability, known as the Monty Hall Problem. Please see any number of YouTube videos on this for an explanation, I'll leave it to them to explain it. I question the predicate of all those who answer this...
I'm taking a look at intuitionistic propositional logic (IPL). Basically it exclude Double Negation Elimination (DNE) from the set of axiom schemas replacing it with Ex falso quodlibet: ⊥ → p for any proposition p (including both atomic and composite propositions). In IPL, for instance, the Law of Excluded Middle (LEM) p ∨ ¬p is no longer a theorem. My question: aside from the logic formal perspective, is IPL supposed to model/address some specific "kind of world" ? Thanks.
Back
Top