(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
Used in a Sentence
bijection
Definition, parts of speech, synonyms, and sentence examples for bijection.
Editorial note
To prove two sets don't have the same cardinality, the most direct way is to prove that such a bijection is impossible.
Examples16
Definitions1
Parts of speech1
Quick take
(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
Meaning at a glance
The clearest senses and uses of bijection gathered in one view.
Definitions
Core meanings and parts of speech for bijection.
noun
(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
Example sentences
To prove two sets don't have the same cardinality, the most direct way is to prove that such a bijection is impossible.
Well my argument is that there is a trivial bijection between the disjoint sets and the parallel lines/points.
They prove division by 3 preserves bijection between arbitrary sets without using the axiom of choice.
A proof that Bool and Bool2 are equal is a bijection f:: Bool -> Bool2.
To me, this bijection makes it clear that you'll get a countable number of pixels with a countable number of doublings.
Technically, you can create a bijection between all natural numbers and all even numbers.
As long as the bijection exists at all, the sets have the same cardinality.
There is a simple bijection between pixels and a subset of rational numbers.
I think you're looking at #3 the wrong way - it's not a bijection, it's meant to be surjective.
If you interpret it into classical logic, it becomes (now in the classical sense) There is no bijection between the natural numbers and the reals.
Then we form a bijection (x, y) <-> x/y and we see that there are as many pixels as rational numbers.
Also, you have to choose the bijection explicitly here because in this case, more than 1 valid one exists, which is another aspect of an isomorphisms 'identity'.
Quote examples
The idea is that if there is any bijection at all between the two sets then they are the same "size".
If "isomorphic" means "in bijection with" and "substructure" means "subset" then this is cardinality.
The proof of "the powerset of the natural is uncountable" (or equivalently, expanding out the definition of uncountability, "there does not exist a bijection between N and P(N)") is constructive, and will hold just as well in intuinistic logic.
I believe that in the context of the infinitude of primes, this is the difference between "NOT (finitely many primes)" and "infinitely many primes", where "infinitely many" means that the primes are in bijection with the natural numbers (and of course finitely many means in bijection with some natural number).
Frequently asked questions
Short answers drawn from the clearest meanings and examples for this word.
How do you use bijection in a sentence?
To prove two sets don't have the same cardinality, the most direct way is to prove that such a bijection is impossible.
What does bijection mean?
(set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
What part of speech is bijection?
bijection is commonly used as noun.