# how does one draw this in tikz?

This is what I've got so far:

\begin{tikzpicture}

\draw[color=blue] (1,0) circle (1.5);
\draw (0,0) node[circle, inner sep=2pt, fill=black, label={above:{$x$}}] (x)  {};
\draw (2,0) node[circle, inner sep=2pt, fill=black, label={above:{$y$}}] (y) {};
\draw [->] (0,0) -- (1.8,0);
\draw [->] (y) .. controls +(-1,-1) .. (0,-0.1);

\end{tikzpicture}


I don't know how to write the equal sign and proceed to the next two pictures.

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One way: You can include the equal sign in a node. Load \usetikzlibrary{positioning}, add \begin{tikzpicture}[node distance=1.5cm] and then \node[left=of y, font=\Huge] {=};, but you could also use chains. – Alenanno Jan 8 at 17:29
For a beginner, ignore all that positioning stuff and just put the \node[font=\huge]{=} where it looks good. If the font isn't big enought, there is \scalebox or \resizebox from the graphicx package. – John Kormylo Jan 8 at 17:49
@JohnKormylo positioning is hard, but \scalebox isn't? :P – Alenanno Jan 8 at 19:05

As @Alenanno said you can just increase size of font with font=\Huge.

You can create style for nodes and arrows to simplify code

Code

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\begin{document}

\tikzset{
Circle/.style={circle, inner sep=2.5pt, fill=black, label={above:{$#1$}}},
Arrows/.style={arrows={-Triangle[length=2mm]}}}

\begin{tikzpicture}[thick,node distance=1cm]
\draw (0,0) node[Circle=x] (x) {};
\draw (2,0) node[Circle=y] (y) {};
\draw [->,Arrows] (x) -- (y);
\draw [->,Arrows] (y) .. controls +(-1,-1)..(x);
\node [right=of y, font=\Huge](z) {\lower1.3ex\hbox{=}};
\node [right=of z, Circle=x](t) {};
\draw [->,Arrows] (t) --+ (2,0)node(e)[label={above:{$y$}}]{};
\node [right=of e, font=\Huge](f) {\lower1.3ex\hbox{$\ast$}};
\node [right=of f,label={above:{$x$}}](g){};
\node [right=2 of g, Circle=y](h){};
\draw [->,Arrows] (h) .. controls +(-1,-1)..(g);
\begin{scope}[color=blue]
\draw (1,0) circle (1.5);
\draw (t)+(1,-1.5) arc (270:90:1.5cm)--cycle;
\draw (h)+(-1,-1.5) arc (-90:90:1.5cm)--cycle;
\end{scope}
\end{tikzpicture}
\end{document}


Output

-

Similarly as in answer of @salimbou, with different way to draw arrows, arc and with use of absolute coordinates:

\documentclass[border=3mm,
tikz,
preview
]{standalone}
\begin{document}
\begin{tikzpicture}[
arrow/.style = {shorten <=1.2mm, shorten >=1.2mm, ->}
]
% 1
\fill[black] (-0.4,0) circle (1mm) node[above] {$x$};
\fill[black] (+0.4,0) circle (1mm) node[above] {$y$};
\draw [arrow] (-0.4,0) -- (0.4,0);
\draw [arrow] (0.4,0) to [out=240,in=300,looseness=2.4] (-0.4,-0);
\draw[color=blue] (0,0) circle (0.8);
% 2
\begin{scope}[xshift=24mm]
\node at (-1.2,-0.07) {\huge$=$};
%
\fill[black] (-0.4,0) circle (1mm) node[above] {$x$};
\node[above] at (+0.4,0) {$y$};
\draw [arrow] (-0.4,0) -- (0.4,0);
\draw[color=blue] (0,0) -- + (0,0.8) arc[start angle=90,end angle=270] -- cycle;
\end{scope}
% 3
\begin{scope}[xshift=44mm]
\node at (-1.1,-0.07) {\huge$*$};
%
\fill[black] (0.4,0) circle (1mm) node[above] {$y$};
\node[above] at (-0.4,0) {$y$};
\draw [arrow] (0.4,0) to [out=240,in=300,looseness=2.4] (-0.4,-0);
\draw[color=blue] (0,0) -- + (0,0.8) arc[start angle=90,end angle=-90] -- cycle;
\end{scope}
\end{tikzpicture}
\end{document}


-
You need \begin{document}. – cfr Jan 9 at 2:04
@cfr, thank you for pointing me on error, its happen because I upload code in two parts ... – Zarko Jan 9 at 2:42