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Let A be a countably infinite set and B = {x,y}. Prove that A x B is countably infinite.

I am not sure what I need to prove here. Is this a disjoint union, I could prove that a disjoint union of any finite set and any countably infinite set is countably infinite. This question is so vague, I am not sure what it is asking me to show. Thank you for any help.

Bret Hisey
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    It seems that you are confused about the Cartesian product $A\times B$. This is notation for the set of ordered pairs $A\times B={(a,b):a\in A,b\in B}$. – Sean English Nov 25 '17 at 03:06
  • Note that $A\times {x,y}$ in this case can be written as the disjoint union of $A\times {x}={(a,x)~:~a\in A}$ and $A\times {y}={(a,y)~:~a\in A}$. For example, if $A$ were $\Bbb N$ you would have $A\times {x}={(1,x),(2,x),(3,x),(4,x),\dots}$. – JMoravitz Nov 25 '17 at 04:10

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Let $\varphi:{\bf{N}}\rightarrow A$ be a bijection, then $\xi:{\bf{N}}\rightarrow A\times B$ defined by $\xi(2k-1)=(\varphi(k),x)$, $\xi(2k)=(\varphi(k),y)$, $k=1,2,...$ is a bijection as well.

Just a note for OP, if you know how to prove that the disjoint union of two countable sets is still countable, you should know the philosophy of my solution. And as @Sean English has noted, that is Cartesian product.

user284331
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  • Downvoted because while this answers the original question, the OP states that they don’t understand what the original question is, so this answer in my opinion is not helpful. – Sean English Nov 25 '17 at 03:08