bijection love

# bijection

## Definitions

### from The American Heritage® Dictionary of the English Language, 4th Edition

• n. Mathematics A function that is both one-to-one and onto.

### from Wiktionary, Creative Commons Attribution/Share-Alike License

• n. A function which is both a surjection and an injection.

## Etymologies

### from The American Heritage® Dictionary of the English Language, 4th Edition

bi-1 + (in)jection or (pro)jection.

### from Wiktionary, Creative Commons Attribution/Share-Alike License

This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

## 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).

Matthew Yglesias » Null Set Blogging

• 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 one-to-one correspondence, between them.)

Slices of Matisse

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

Dedekind's Contributions to the Foundations of Mathematics

• On the other hand, in Mirimanoff's 1917a there is a remarkable use of Burali-Forti's paradox which suggests a necessary condition for set-hood 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.

Paradoxes and Contemporary Logic

• The bijection we have just observed can now be stated as

Algebra

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

Facts

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

Facts