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I hope I have not hereby created a duplicate, please perdon me if I did, but I had this question for a while now:

Let $A \& B $ be two sets such that $A \subseteq B$. Suppose there exist a one to one (bijective) function $f : A \to B $. Then have we got $|A| = |B|$?

I know that if these sets are finite, it works, but what about the infinite case

Thank you

T. D

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  • $\begingroup$ What you've written is false, but if you change $A=B$ to $|A|=|B|$, it is true. It is also true for the infinite case by the definition of cardinality. $\endgroup$ – Kevin Long Nov 18 '19 at 19:15
  • $\begingroup$ That is what I meant, $\endgroup$ – T.D. Nov 18 '19 at 19:16
  • $\begingroup$ It's always true that if there is a bijection $A \to B$ then $|A| = |B|$, regardless of whether $A$ is a subset of $B$. $\endgroup$ – Clive Newstead Nov 18 '19 at 19:30
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Let $A = 2\mathbb{Z}$ (ie the even integers) and $B = \mathbb{Z}$. Then, $A \subseteq B$, and there is a bijection between them, (namely $f:B \to A$ defined by $f(x) = 2x$) but the sets are not equal

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  • $\begingroup$ I did a mistake see the corrected $\endgroup$ – T.D. Nov 18 '19 at 19:17
  • $\begingroup$ Well, by definition of cardinality, if there exists a bijection between the sets, then they have the same cardinality. $\endgroup$ – Noah Caplinger Nov 18 '19 at 19:34

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