FACTOID # 53: If you thought Antarctica was inhospitable, think again - its land area is only ninety-eight percent ice. Reassuringly, the other 2% is categorised as "barren rock".
 
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Encyclopedia > Equinumerosity

Two sets A and B are said to be equinumerous if they have the same cardinality, i.e., if there exists a bijection f : A → B. This is denoted

The study of cardinality is often called equinumerosity. Sometimes the terms equipollent or even equivalent are used, though equivalent is a bit overused. More at cardinality.


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Frege's Logic, Theorem, and Foundations for Arithmetic > Frege's Derivation of Hume's Principle in the Grundlagen ... (514 words)
Then, by the definitions of ‘the number of Ps’ and ‘the number of Qs’, we know that the extension of the concept equinumerous to P is identical with the extension of the concept equinumerous to Q. But it is a fact about equinumerosity that P is equinumerous to P.
So by the transitivity of equinumerosity (this is Fact 4 in the subsection on Equinumerosity in the main portion of the entry), S is equinumerous to Q.
By the symmetry of equinumerosity (Fact 3 in the subsection on Equinumerosity), it follows that Q is equinumerous to S.
mmtheorems41 - Metamath Proof Explorer (451 words)
The domain of a one-to-one function is dominated by its range.
Equinumerosity inference from an implicit one-to-one onto function.
A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its range.
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