[Tex/LaTex] Change existing symbol size

macros

I would like to make the \circ symbol smaller and centered when using it between two letters. For example, I want to say

\begin{align}
\forall\  a,b,c \in S,\ a\circ( b\circ c)= (a\circ b)\circ c
\end{align}

The default symbol is just too big for my tastes. Please bear in mind that I am a newbie.

Edit: My preamble.

\documentclass[letterpaper,12pt]{memoir}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{braket}
\usepackage{graphicx}
\usepackage{fancyvrb}
setlrmarginsandblock{1.25in}{1.0in}{1}
\setulmarginsandblock{1in}{1in}{1}
\checkandfixthelayout

\theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{proposition}  {Proposition} \newtheorem{lemma}{Lemma} \newtheorem*{corollary}{Corollary}
\theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem{conjecture}   {Conjecture} \newtheorem*{example}{Example} \newtheorem{algorithm}{Algorithm}
\theoremstyle{remark} \newtheorem*{remark}{Remark} \newtheorem*{note}{Note}      \newtheorem{case}{Case}
\theoremstyle{algorithm} \newtheorem*{properties}{Properties}

\mathchardef\ordinarycolon\mathcode`\:
\mathcode`\:=\string"8000
\begingroup \catcode`\:=\active
\gdef:{\mathrel{\mathop\ordinarycolon}}
\endgroup

\begin{document}
\include{front_cover}
\tableofcontents
\include{lecture_3}
\end{document}

Best Answer

I'm usually pretty satisfied with the size of \circ. But if you want it reduced you can do like this

\documentclass{article}
\newcommand{\ccirc}{\mathbin{\mathchoice
  {\xcirc\scriptstyle}
  {\xcirc\scriptstyle}
  {\xcirc\scriptscriptstyle}
  {\xcirc\scriptscriptstyle}
}}
\newcommand{\xcirc}[1]{\vcenter{\hbox{$#1\circ$}}}
\begin{document}
$f\circ g$

$f\ccirc g$

$\displaystyle\lim_{f\ccirc g}$
\end{document}

The first line is just for comparison.

enter image description here

How to "exchange" the \circ command with the new one? Just replace the given code with

\let\latexcirc=\circ
\newcommand{\ccirc}{\mathbin{\mathchoice
  {\xcirc\scriptstyle}
  {\xcirc\scriptstyle}
  {\xcirc\scriptscriptstyle}
  {\xcirc\scriptscriptstyle}
}}
\newcommand{\xcirc}[1]{\vcenter{\hbox{$#1\latexcirc$}}}
\let\circ\ccirc

You need to comment out only the last line in case you want to return back to the default.

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