XIV
The "isomorphism" notion
This is done with the "isomorphism" concept. The idea is very simple:
We say that two objects
( X L , 6 I )
and (^2,62) from a mathematical field
are isomorphic if there exists a one-to-one surjective (i.e., onto) function
f:Xi X2 (called an isomorphism) such that /(@i) = ©2- The last
identity should be intrepreted in the sense that / "preserves" the structures
61 and &2 of the sets X\ and X2, respectively. For instance, if X\ and X2
are isomorphic vector spaces via the isomorphism / , then (besides / being
one-to-one and onto) it also satisfies f(ax -\- (3y) otf{x) + (3f(y) for all
x j G l i and all scalars a and (3. Likewise, by saying that two ordered
vector spaces X\ and X2 are isomorphic via / , we mean that / : X\ X2
is a one-to-one surjective (linear) operator such that f(x) f(y) holds in
X2 if and only if x y holds in X\.
The word "isomorphism" is the English version of the Greek word
LiiaoiAOp(j)Lcrn6s"
which etymologically is the composition of the Greek
words
"LCTOS"
(which means equal, even, the same) and "fjopfiri" (which
means form, figure, shape, appearance, structure). So, when we say that
two mathematical objects are "isomorphic," we simply express the fact that
they have the same (or similar or identical) shape (or form or appearance)
and the concept of an "isomorphism" simply designates the state of being
"isomorphic."
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