Gsyfuy

Can someone show me how to shade the yellow region by modifying my code?
Thank you in advance guys :)

documentclass[]{article}

usepackage[margin=0.5in]{geometry}

usepackage{pgfplots}

renewcommand{thesection}{arabic{section}}

usepackage{mathtools}

usepackage{cancel}

usepackage{pgfplots}

usepackage{amsmath}

newtheorem{theorem}{THEOREM}

newtheorem{proof}{PROOF}

usepackage{tikz}

usepackage{amssymb}

usetikzlibrary{patterns}

usepackage{fancyhdr}

usepackage{bigints}

usetikzlibrary{angles}

usepackage{tcolorbox}

usepackage{color,xcolor}

usepackage{booktabs,array}

usepackage{hyperref}

usepackage{graphicx}

usetikzlibrary{arrows}

usepackage{polynom}

usepackage{wallpaper}

usetikzlibrary{shapes.geometric}

usepgfplotslibrary{fillbetween}

newenvironment{tightcenter}{

setlengthtopsep{0pt}

setlengthparskip{0pt}

begin{center}}{end{center}}

begin{document}

begin{tikzpicture}

draw[thick] (-3,-3.9) rectangle (4,2);

draw[thick] (0,0) circle (1.5cm);

draw[thick] (1.5,0) circle (1.5cm);

draw[thick] (0.75,-1.6) circle (1.5cm);

%

%scope

%clip (-3,-2) rectangle (4,2)

%(0,0) circle (1.5cm);

%fill[gray] (1.5,0) circle (1.5cm);

%endscope

%

node [above] at (-3.1,2){$mathcal{E}$};

node at (-1.2,1.5) {$A$};

node at (+2.6,1.5) {$B$};

node at (+0.75,-3.4) {$C$};

end{tikzpicture}

end{document}

Straightforward with reverseclip.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{backgrounds}

% based on https://tex.stackexchange.com/a/12033/121799

tikzset{reverseclip/.style={insert path={(current bounding box.south west)rectangle 

(current bounding box.north east)} }} 

begin{document}

begin{tikzpicture}

draw[thick] (-3,-3.9) rectangle (4,2);

draw[thick] (0,0) circle (1.5cm);

draw[thick] (1.5,0) circle (1.5cm);

draw[thick] (0.75,-1.6) circle (1.5cm);

node [above] at (-3.1,2){$mathcal{E}$};

node at (-1.2,1.5) {$A$};

node at (+2.6,1.5) {$B$};

node at (+0.75,-3.4) {$C$};

begin{scope}[on background layer]

clip (0,0) circle (1.5cm) [reverseclip];

clip (1.5,0) circle (1.5cm);

fill[yellow] (0.75,-1.6) circle (1.5cm);

end{scope}

end{tikzpicture}

end{document}

One can of course avoid repetition and make it a bit more automatic by storing the paths in macros and using pgf keys. Then it boils down to

begin{scope}[on background layer]

pgfkeys{not inside=pathA,inside/.list={pathB,pathC},shade=yellow}

end{scope} 

with the same output.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{backgrounds}

% based on https://tex.stackexchange.com/a/12033/121799

tikzset{reverseclip/.style={insert path={(current bounding box.south west)rectangle 

(current bounding box.north east)} }} 

makeatletter % https://tex.stackexchange.com/a/38995/121799

tikzset{

  use path/.code={pgfsyssoftpath@setcurrentpath{#1}}

}

makeatother

defPATH{path}

begin{document}

pgfkeys{not inside/.code={clip[use path=#1,reverseclip];},

inside/.code={clip[use path=#1];},

shade/.code=fill[#1] (current bounding box.south west)rectangle 

(current bounding box.north east);}

begin{tikzpicture}

draw[thick] (-3,-3.9) rectangle (4,2);

draw[thick,save path=pathA] (0,0) circle (1.5cm);

draw[thick,save path=pathB] (1.5,0) circle (1.5cm);

draw[thick,save path=pathC] (0.75,-1.6) circle (1.5cm);

node [above] at (-3.1,2){$mathcal{E}$};

node at (-1.2,1.5) {$A$};

node at (+2.6,1.5) {$B$};

node at (+0.75,-3.4) {$C$};

begin{scope}[on background layer]

pgfkeys{not inside=pathA,inside/.list={pathB,pathC},shade=yellow}

end{scope}

end{tikzpicture}

end{document}