Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
- noun A function that is both one-to-one and onto.
from Wiktionary, Creative Commons Attribution/Share-Alike License.
- noun set theory A function which is both a
surjection and aninjection .
Etymologies
from The American Heritage® Dictionary of the English Language, 4th Edition
from Wiktionary, Creative Commons Attribution/Share-Alike License
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Examples
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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).
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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.
Skolem's Paradox Bays, Timothy 2009
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Ω (mË) says that there is no bijection between the natural numbers and mË.
Skolem's Paradox Bays, Timothy 2009
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(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 gerard varni 2009
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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
Skolem's Paradox Bays, Timothy 2009
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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 Reck, Erich 2008
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The bijection we have just observed can now be stated as
Algebra Pratt, Vaughan 2007
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F is a bijection from W onto the set of all maximal consistent sets of facts.
Facts Mulligan, Kevin 2007
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F is a bijection from W onto the set of all conjunctively complete sets of facts;
Facts Mulligan, Kevin 2007
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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 Cantini, Andrea 2007
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