# Schroeder–Bernstein theorem

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2020 Mathematics Subject Classification: Primary: 03E20 [MSN][ZBL]

Cantor–Bernstein theorem

For sets \$A\$ and \$B\$, if there are injections from \$A\$ to \$B\$ and from \$B\$ to \$A\$ (equivalently, each is equipotent to a subset of the other), then there is a bijection between \$A\$ and \$B\$ (they are equipotent sets).

In cardinal arithmetic, if we let \$\mathfrak{a} \le \mathfrak{b}\$ denote the property that some set of cardinality \$\mathfrak{a}\$ has an injection to a set of cardinality \$\mathfrak{b}\$, then \$\mathfrak{a} \le \mathfrak{b}\$ and \$\mathfrak{b} \le \mathfrak{a}\$ implies \$\mathfrak{a} = \mathfrak{b}\$.

The theorem was conjectured by Georg Cantor by 1895 and proved by Felix Bernstein in 1897. Dedekind obtained a further proof in 1897. Schroeder's proof of 1898 was found to be flawed by 1902.

#### References

• P. R. Halmos, "Naive Set Theory", Springer (1960) ISBN 0-387-90092-6
• Michael Potter, "Set Theory and its Philosophy : A Critical Introduction", Oxford University Press (2004) ISBN 0-19-155643-2
How to Cite This Entry:
Schroeder–Bernstein theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schroeder%E2%80%93Bernstein_theorem&oldid=54511