The other is to construct its inverse explicitly, thereby showing that it has an inverse and hence that it must be a bijection. You could take that approach to this problem as well:

Oct 24, 2010 · I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural …

May 21, 2025 · 44 "Same cardinality" is defined as meaning there is a bijection. In your vector space example, you were requiring the bijection to be linear. If there is a linear bijection, the dimension is …

Apr 15, 2020 · 2 A bijection is an isomorphism in the category of Sets. When the word "isomorphism" is used, it is always referred to the category you are working in. I will list some categories including their …

If you only have to show that such bijection exists, you can use Cantor-Bernstein theorem and $ (0,1)\subseteq (0,1] \subseteq (0,2)$. See also open and closed intervals have the same cardinality …

Oct 9, 2019 · No, you can't always find a bijection between two uncountable sets. For example, there is never a bijection between any set and its powerset (and sorry, but the standard proof is …

Sep 7, 2017 · A bijection, being a mapping, is usually depicted with one-directional arrows or rays relating the elements. In the former case the distinction between the domain and range is not really …

Jan 2, 2025 · I am having some trouble with an intuitive understanding of how we can say two sets equal in cardinality iff there is a bijection between them. In particular, a bijection exists between …

Jan 21, 2025 · To my understanding, an isomorphism is a bijection that also preserves a specific structure, such as algebraic or geometric operations. While every isomorphism is a bijection, not all …

Jan 6, 2024 · Condition 3 is necessary, but not sufficient: a linear map between two spaces of different dimensions cannot be bijective. Since often one knows the dimensions ahead of time, this is an …