From e1407b2eabe7e4c5fcf8e899915817a70a3e2192 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Hillerstr=C3=B6m?= Date: Wed, 21 Oct 2020 01:36:14 +0100 Subject: [PATCH] Slight elaboration --- thesis.tex | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/thesis.tex b/thesis.tex index 744bf2e..e829706 100644 --- a/thesis.tex +++ b/thesis.tex @@ -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. % \[ - \dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \} + \dec{Im}(f) \defas \{\, f(a) \mid a \in \dom(f) \}. \] @@ -431,6 +431,12 @@ written $\dec{Im}(f)$, is the set of values that it can return, i.e. to be a bijective. \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 whenever the function $f' : \dom(f) \to \dec{cod}(f)$, obtained by restricting $f$ to its domain, is injective, surjective, and bijective