Assertion: isomorphism | /r/HasProperty | bijection
Facets:
value: on object sets
type: LOCATION
value: in graph theory
type: OTHER
Source sentences:
text: In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H
text: In graph theory, an isomorphism of graphs $G$ and $H$ is a bijection between the vertex sets of $G$ and $H$, $ f \colon V(G)
text: Thus, in every category of algebraic systems, an isomorphism is a homomorphism that is a bijection.
text: If the categories are small (having sets of objects and morphisms) then an isomorphism of categories is a bijection on object sets and morphism sets.
text: It's true that an isomorphism in a concrete category must be a bijection on underlying sets; but it's possible for the objects of a category to admit something we might call an underlying set without being a concrete category, and then an isomorphism might not be bijective.
text: Because an isomorphism is a bijection, every element of $G'$ has exactly one preimage; in particular there is exactly one preimage of $e_{G'}$.
text: In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H
text: H\) is an isomorphism of graphs, than we must have \(d(\varphi(v))=d(v)\) for all vertices \(v\in G\text{,}\) and since isomorphisms are bijections on the vertex set, we see the degree sequence must be preserved.