How to define a bijection between $(0,1)$ and $(0,1]$? If A is a countable infinite set and a is any element in A, then: there's a bijection for A \ {a} and A; Or, equivalently if A is contained strictly in B, b is in B \ A, A is countably infinite then: there's a bijection for A $\bigcup$ {b} and A;
is there a bijection for $f: \\mathbb R \\to \\mathbb C$? If you just want a bijection as sets, the answer is yes That’s exactly what it means for two sets to have the same cardinality If you’re looking for a bijection that preserves some structure, that will depend on precisely what you’re trying to preserve but the answer is very probably no
elementary set theory - Bijection and Uncountable Sets (understanding . . . we can find a bijection between any two countable sets (I think this is correct) No For example, some countable sets are countably infinite while others are finite However, there is a bijection between any two countably infinite sets If we find a bijection between two finite sets, then the two sets must be of the same cardinality Yes
functions - Notation for bijection - Mathematics Stack Exchange That is, given $\sigma\in S_3$ we get a bijection $\alpha\circ\sigma\circ\chi^{-1}\colon X\to A$ and any bijection can be produced in such a way Now take your favourite notation for peremutations Share
Bijective vs Isomorphism - Mathematics Stack Exchange 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 typical names for isomorphism: Sets: Bijection; Groups: Isomorphism; Top: Homeomorphism; Differentiable Manifolds: Diffeomorphism; Riemannian Manifolds