I would definitely go the second way with $\displaystyle: I find that it looks strange to have something centered after a bullet. To get the vertical alignment correct, just add [t] after \begin{aligned}, where "t" means "top". (See also the top of page 8 of the User's guide for the amsmath Package.)
The immediate cause of the error you're getting would appear to be the \\* snippet; I can't tell what it's for. LaTeX will also choke on \inX, \subseteqX, and \subseteqY; there needs to a be a gap between the control sequence and the subsequent material. The use of \item statements while not in a list environment (such as itemize or enumerate) is also going to throw an error.
I would strongly recommend using \mid rather than \vert when expressing thoughts such as "conditional on", "given that", and "subject to". The macro \mid adds some horizontal spacing around the vertical bar, whereas \vert does not. Knuth, by the way, also recommends inserting \, (a "thin space") after the opening left curly brace and before the closing right curly brace in the conditioning statement.
Since you posted a code snippet rather than an entire MWE (minimum working example), I had to make a guess as to how the definition environment is defined.
\documentclass{article}
\usepackage{amsmath,amsthm}
\theoremstyle{definition} % adjust this setup to your needs
\newtheorem{definition}{Definition}
\newcommand{\image}{\mathit{im}} % better to define a macro to define the appearance of "image"
\begin{document}\pagestyle{empty}
\begin{definition}[1.08]
Suppose $f : X\longrightarrow Y$ is a function, and $A\subseteq X$, $B\subseteq Y$.
\begin{itemize}
\item[(i)] We define the image set of $f$, denoted $\image(f)$, by $\image(f)=\{\,f(x) \mid x\in X\,\}$.
\item[(ii)] We define the image set of $A$ under $f$, denoted $f(A)$, by $f(A)=\{\,f(x) \mid x\in A\,\}$.
\item[(iii)] We define the inverse image set of $B$ under $f$, denoted $f^{-1}(B)$, by $f^{-1}(B)=\{\,x\in X \mid f(x)\in B\,\}$.
\end{itemize}
\end{definition}
\end{document}
Best Answer
The main issue is that you mustn't have blank lines in an
equationenvironment -- or any display-math environment, for that matter.An additional concern: If you use Times Roman as the text font, you should probably also load a Times Roman math font. The
mathptmxandnewtxtext/newtxmathpackages may be suitable.