Understanding Isomorphic Across Disciplines | Generated by AI
Question: What does “isomorphic” mean?
Answer:
Isomorphic means “having the same shape/structure/form” — two things are isomorphic if they are structurally identical, even if they look different on the surface.
The precise meaning depends on the field:
In mathematics (most common usage)
Two objects are isomorphic if there exists a bijective (one-to-one and onto) mapping between them that preserves all relevant structure.
Examples by field:
-
Graph theory
Two graphs are isomorphic if you can relabel the vertices of one graph so that it becomes exactly the same as the other (same connections). -
Group theory
Two groups (G, ⋅) and (H, ∗) are isomorphic if there is a bijection φ: G → H such that
φ(a ⋅ b) = φ(a) ∗ φ(b) for all a, b ∈ G
(they have exactly the same multiplication table, just with different names for elements) -
Linear algebra
Two vector spaces are isomorphic if they have the same dimension (there exists a linear bijection between them). -
Category theory
Two objects are isomorphic if there is an isomorphism (a morphism that has an inverse morphism) between them.
Quick summary table:
| Field | What “same structure” means | Typical isomorphism symbol |
|---|---|---|
| Sets | Same cardinality (bijection exists) | ≈ or ≅ |
| Graphs | Same adjacency relation after relabeling | ≅ |
| Groups | Same operation table | ≅ |
| Vector spaces | Same dimension | ≅ |
| Rings / Fields | Preserve addition and multiplication | ≅ |
| Topological spaces | Homeomorphism (continuous bijection with continuous inverse) | ≈ or ≅ |
In other contexts (less formal)
- Chemistry: molecules are isomorphic if they have identical connectivity (sometimes called “isostructural”).
- Programming / databases: two data structures / schemas are isomorphic if they can represent exactly the same information with a lossless conversion.
- Everyday language (rare): “structurally the same” or “equivalent in form”.
Most common correct short answer in math/CS:
“Isomorphic = there is a structure-preserving bijection between them.”
References:
- None (this is standard mathematical terminology)