Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. Mathematics A function that is both onetoone and onto.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. A function which is both a surjection and an injection.
Etymologies
Examples

You can make each point have measure 1/2, for example, or assign whatever positive weights you like to each point (such measures are in bijection with functions from your base set to the positive reals).

Î© (mË) says that there is no bijection between the natural numbers and mË.

If asked about the phrase “is a bijection,” she will go on to talk about collections of ordered pairs satisfying certain nice properties, and if asked about the term

If we are willing make the further assumption that it only takes one bijection to one such instance of the power set of Ï to render the power set itself “absolutely” countable, then we can understand the Skolemite's strong claim about absolute countability.

(A set is reflexive iff it is equipollent to one of its proper subsets; and two sets are equipollent with one another iff there exists a bijection, i.e., a onetoone correspondence, between them.)

Dedekind also provided a proof of the CantorBernstein Theorem (that between any two sets which can be embedded into each other onetoone there exists a bijection, so that they have the same cardinality), another basic result in the modern theory of transfinite cardinals.

On the other hand, in Mirimanoff's 1917a there is a remarkable use of BuraliForti's paradox which suggests a necessary condition for sethood in terms of size, viz., if a collection is in bijection with the set of all ordinals, then it does not exist as a set.

The bijection we have just observed can now be stated as

F is a bijection from W onto the set of all maximal consistent sets of facts.

F is a bijection from W onto the set of all conjunctively complete sets of facts;
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