Mryntu

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
usepackagepgfplots
renewcommandthesectionarabicsection
usepackagemathtools
usepackagecancel
usepackagepgfplots
usepackageamsmath
newtheoremtheoremTHEOREM
newtheoremproofPROOF
usepackagetikz
usepackageamssymb
usetikzlibrarypatterns
usepackagefancyhdr
usepackagebigints
usetikzlibraryangles
usepackagetcolorbox
usepackagecolor,xcolor
usepackagebooktabs,array
usepackagehyperref
usepackagegraphicx
usetikzlibraryarrows
usepackagepolynom
usepackagewallpaper
usetikzlibraryshapes.geometric
usepgfplotslibraryfillbetween
newenvironmenttightcenter
setlengthtopsep0pt
setlengthparskip0pt
begincenterendcenter
begindocument
begintikzpicture
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)$mathcalE$;
node at (-1.2,1.5) $A$;
node at (+2.6,1.5) $B$;
node at (+0.75,-3.4) $C$;
endtikzpicture
enddocument

Straightforward with reverseclip.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarybackgrounds
% based on https://tex.stackexchange.com/a/12033/121799
tikzsetreverseclip/.style=insert path=(current bounding box.south west)rectangle 
(current bounding box.north east) 
begindocument
begintikzpicture
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)$mathcalE$;
node at (-1.2,1.5) $A$;
node at (+2.6,1.5) $B$;
node at (+0.75,-3.4) $C$;
beginscope[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);
endscope
endtikzpicture
enddocument

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

beginscope[on background layer]
pgfkeysnot inside=pathA,inside/.list=pathB,pathC,shade=yellow
endscope 

with the same output.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarybackgrounds
% based on https://tex.stackexchange.com/a/12033/121799
tikzsetreverseclip/.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
defPATHpath
begindocument
pgfkeysnot 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);
begintikzpicture
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)$mathcalE$;
node at (-1.2,1.5) $A$;
node at (+2.6,1.5) $B$;
node at (+0.75,-3.4) $C$;
beginscope[on background layer]
pgfkeysnot inside=pathA,inside/.list=pathB,pathC,shade=yellow
endscope
endtikzpicture
enddocument