It should be mentioned that it isn't always true for infinite sets. E.g., take ##A = B = \mathbb{R}## and ##X = \mathbb{Z}##. In general it will be possible only if ##X## has the same cardinality as ##B## (and hence the same cardinality as ##A##).UltrafastPED said:I asked a question; I did not provide an answer.
However it seems that you now know the answer: I think you are saying "for finite sets the answer is NO; for infinite sets the answer is YES". Next you should go ahead and construct an example using the set of natural numbers and subsets of odds and evens.
A bijection is a type of function in mathematics where every element in the domain (A) is paired with a unique element in the codomain (B). This means that every element in A has one and only one corresponding element in B, and vice versa.
To determine if a function is a bijection, you must check two things: injectivity and surjectivity. A function is injective if each element in the codomain (B) has at most one corresponding element in the domain (A). A function is surjective if every element in the codomain (B) has at least one corresponding element in the domain (A). If a function satisfies both of these conditions, it is a bijection.
Bijections are important in mathematics because they provide a one-to-one correspondence between two sets. This allows for the manipulation and comparison of elements in the two sets, making it easier to solve problems and prove theorems.
Yes, a function can still be a bijection even if the domain and codomain are not equal. As long as every element in the domain has a unique corresponding element in the codomain, and vice versa, the function is considered a bijection.
The inverse of a bijection is a function that reverses the mapping of the original bijection. This means that the domain and codomain switch places, and the direction of the mapping is reversed. The inverse of a bijection is also a bijection, as every element in the new domain has a unique corresponding element in the new codomain, and vice versa.