Proving that uncountable sets are of the same cardinality

Asked May 05 '20 at 08:26

Active May 05 '20 at 09:43

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In this week's discrete mathematics' class, we received the following 2 exercise subsections:

prove that the sets: $[a,b],(a,b)$ have similar cardinalities.
prove that the sets: $(a,b),(0,1)$ have similar cardinalities.
if we know that $A=\{x\in \mathbb N,2|x\},B=\{x\in \mathbb N, 3|x\},$ prove that $C=A\cup B$ has the same cardinality.

I'm lost, especially with the 2 first questions. In the third one, I have succeded in proving that $A$ and $B$ have the same cardinality, but can't seem to find a bijective function from $A$ to $C$.

edited May 05 '20 at 09:39

asked May 05 '20 at 08:26

C.H.

In the first one, you can start by showing that $[a,b] = (a,b)\cup {a}\cup {b}$. – Alejandro Bergasa Alonso May 05 '20 at 08:31
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There's no such thing as "similar cardinalities". Cardinalities of two sets are either the same or different. – Clement Yung May 05 '20 at 08:33
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Can you be a bit more specific about your problems? – Kuhlambo May 05 '20 at 08:34
@ClementYung I guess it means equal cardinalities. – Alejandro Bergasa Alonso May 05 '20 at 08:45
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Currently, $C$ has a cardinality of $1$. You probably didn't mean to add the parentheses { }. – Linear Christmas May 05 '20 at 08:50
Fixed, and indeed I meant equal cardinalities. – C.H. May 05 '20 at 09:39
There are probably several other duplicates as well. Please search before posting your questions, and please keep them focused. – Asaf Karagila May 05 '20 at 09:43

Proving that uncountable sets are of the same cardinality

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