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

  • $\begingroup$ 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

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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.