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Slight elaboration

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Daniel Hillerström
2020-10-21 01:36:14 +01:00
parent 16aff4f445
commit e1407b2eab
1 changed files with 7 additions and 1 deletions
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thesis.tex
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@@ -416,7 +416,7 @@ is that of \emph{image}. The image of a total or partial function $f$,
written $\dec{Im}(f)$, is the set of values that it can return, i.e. written $\dec{Im}(f)$, is the set of values that it can return, i.e.
% %
\[ \[
\dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \} \dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \}.
\] \]
@@ -431,6 +431,12 @@ written $\dec{Im}(f)$, is the set of values that it can return, i.e.
to be a bijective. to be a bijective.
\end{definition} \end{definition}
% %
An injective function guarantees that each element in its image is
uniquely determined by some element of its domain.
%
A surjective function guarantees that its domain covers the codomain,
meaning that the codomain and image coincide.
%
A partial function $f$ is injective, surjective, and bijective A partial function $f$ is injective, surjective, and bijective
whenever the function $f' : \dom(f) \to \dec{cod}(f)$, obtained by whenever the function $f' : \dom(f) \to \dec{cod}(f)$, obtained by
restricting $f$ to its domain, is injective, surjective, and bijective restricting $f$ to its domain, is injective, surjective, and bijective