After reading this Math.SE answer, I wanted to know if my understanding of partial vs. total functions is correct.
$$ \newcommand{\+}{\mkern2mu} \newcommand{\viff}{\Big\Updownarrow} \begin{gather*} f \subseteq A \times B \text{ is a }\textit{partial} \text{ function} \\[5pt] \viff \\[5pt] \forall\+ a \in A,\, \forall\+ b \in B,\, \forall\+ c \in B,\; \bigl((a, b) \in f \land (a, c) \in f\+\bigr) \implies b = c. \\[10pt] f \subseteq A \times B \text{ is a } \textit{total} \text{ function} \\[5pt] \viff \\[5pt] \Bigl(\+\forall\+ a \in A,\, \forall\+ b \in B,\, \forall\+ c \in B,\; \bigl((a, b) \in f \+\land\+ (a, c) \in f\+\bigr) \implies b = c.\Bigr) \\[5pt] \land \\[5pt] \bigl(\+\forall\+ x \in A,\, \exists\, y \in B,\; (x, y) \in f.\bigr) \end{gather*} $$
In other words, $$ \begin{aligned} f \text{ is a } \textit{partial} \text{ function} &\iff f \text{ is a univalent relation.} \\[5pt] f \text{ is a } \textit{total} \text{ function} &\iff f \text{ is a total and univalent relation.} \\[5pt] \end{aligned} $$
Equivalently(?),
$$ \begin{gather*} f\+\colon\+ A \to B\+ \text{ is a } \textit{partial} \text{ function} \\[5pt] \viff \\[5pt] \forall\+ a \in A,\, \forall\+ b \in B,\, \forall\+ c \in B,\; \bigl(f(a) = b \+\land\+ f(a) = c\bigr) \implies b = c. \\[10pt] f\+\colon\+ A \to B\+ \text{ is a } \textit{total} \text{ function} \\[5pt] \viff \\[5pt] \Bigl(\+\forall\+ a \in A,\, \forall\+ b \in B,\, \forall\+ c \in B,\; \bigl(f(a) = b \+\land\+ f(a) = c\+\bigr) \implies b = c.\Bigr) \\[5pt] \land \\[5pt] \bigl(\+\forall\+ x \in A,\, \exists\, y \in B,\; f(x) = y.\bigr) \end{gather*} $$
Are these definitions correct?
Also, when is it safe to assume that function refers to total function in mathematical texts?
Thank you.
