Vrftjkry

In the next code, I display the contents of two sets A and B respectively, and the arrows that determine the mapping between their elements...

What is left, is to draw the two sets (namely, two ellipses - one in the left that encloses ai's and xi's and one in the right that encloses the bi's and the yi's). The names of the two sets A and B have to be displayed after the bottom of the two ellipses. How can I do this?

documentclass[a4paper,twoside,10pt]{book}

usepackage {tikz}

usetikzlibrary {positioning,graphs,calc,decorations.pathmorphing,shapes,arrows.meta,arrows,shapes.misc}

usepackage{tikz-cd}

usepackage{greektex}

usepackage[american,greek,greek.polytoniko]{babel}





begin{document}



begin{center}

tikzset{every loop/.style={min distance=15mm,looseness=10}}

begin{tikzpicture}[-latex ,auto ,node distance =0.7cm and 5cm, on grid,semithick ,

state/.style ={circle, draw, color=blue , fill=blue, text=white , minimum width =0.2 cm}]

node[state] (a1) [label=left:$alpha_1$]{};

node[state] (a2) [below =of a1][label=left:$alpha_2$]{};

node[state] (a3) [below =of a2][label=left:$alpha_3$]{};

node[state] (adots) [below =of a3][label=left:$vdots$]{};

node[state] (an) [below =of adots][label=left:$alpha_n$]{};

node[state] (b1) [right =of a1][label=right:$beta_1$]{};

node[state] (b2) [below =of b1][label=right:$beta_2$]{};

node[state] (b3) [below =of b2][label=right:$beta_3$]{};

node[state] (bdots) [below =of b3][label=right:$vdots$]{};

node[state] (bn) [below =of bdots][label=right:$beta_n$]{};

node[state] (x1) [below =of an,yshift=-0.5cm][label=left:$x_1$]{};

node[state] (x2) [below =of x1][label=left:$x_2$]{};

node[state] (x3) [below =of x2][label=left:$x_3$]{};

node[state] (xdots) [below =of x3][label=left:$vdots$]{};

node[state] (xn) [below =of xdots][label=left:$x_n$]{};

node[state] (y1) [below =of bn,yshift=-0.5cm][label=right:$y_1$]{};

node[state] (y2) [below =of y1][label=right:$y_2$]{};

node[state] (y3) [below =of y2][label=right:$y_3$]{};

node[state] (ydots) [below =of y3][label=right:$vdots$]{};

node[state] (yn) [below =of ydots][label=right:$y_n$]{};

path [dashed] (a1) edge (b1); path [dashed] (b2) edge (a2);

path [dashed] (b3) edge (a3); path [dashed] (an) edge (bn);

path (x1) edge (y1); path (y2) edge (x2);

path (y3) edge (x3); path (xn) edge (yn);

end{tikzpicture}

end{center}

end{document} 

In addition to @DungVu's answer, it is possible to use the fit library, which allows you to create a node that includes all the others. Here, as they are aligned, you just have to include one from above and below.
I chose the second (a2) and second last (xdots) to have a prettier ellipse for my taste.

node[fit={(a2)(xdots)},draw, ellipse,minimum width=3cm](left){};

documentclass[a4paper,twoside,10pt]{book}

usepackage {tikz}

usetikzlibrary {positioning,graphs,calc,decorations.pathmorphing,shapes,arrows.meta,arrows,shapes.misc}

usetikzlibrary{fit}

%usepackage{tikz-cd}

%usepackage{greektex}

%usepackage[american,greek,greek.polytoniko]{babel}



begin{document}



begin{center}

tikzset{every loop/.style={min distance=15mm,looseness=10}}

begin{tikzpicture}[-latex ,auto ,node distance =0.7cm and 5cm, on grid,semithick ,

state/.style ={circle, draw, color=blue , fill=blue, text=white , minimum width =0.2 cm}]

node[state] (a1) [label=left:$alpha_1$]{};

node[state] (a2) [below =of a1][label=left:$alpha_2$]{};

node[state] (a3) [below =of a2][label=left:$alpha_3$]{};

node[state] (adots) [below =of a3][label=left:$vdots$]{};

node[state] (an) [below =of adots][label=left:$alpha_n$]{};

node[state] (x1) [below =of an,yshift=-0.5cm][label=left:$x_1$]{};

node[state] (x2) [below =of x1][label=left:$x_2$]{};

node[state] (x3) [below =of x2][label=left:$x_3$]{};

node[state] (xdots) [below =of x3][label=left:$vdots$]{};

node[state] (xn) [below =of xdots][label=left:$x_n$]{};

% left ellipse

node[fit={(a2)(xdots)},draw, ellipse,minimum width=3cm](left){};



node[state] (b1) [right =of a1][label=right:$beta_1$]{};

node[state] (b2) [below =of b1][label=right:$beta_2$]{};

node[state] (b3) [below =of b2][label=right:$beta_3$]{};

node[state] (bdots) [below =of b3][label=right:$vdots$]{};

node[state] (bn) [below =of bdots][label=right:$beta_n$]{};

node[state] (y1) [below =of bn,yshift=-0.5cm][label=right:$y_1$]{};

node[state] (y2) [below =of y1][label=right:$y_2$]{};

node[state] (y3) [below =of y2][label=right:$y_3$]{};

node[state] (ydots) [below =of y3][label=right:$vdots$]{};

node[state] (yn) [below =of ydots][label=right:$y_n$]{};

% right ellipse

node[fit={(b2)(ydots)},draw, ellipse,minimum width=3cm](right){};



path [dashed] (a1) edge (b1); path [dashed] (b2) edge (a2);

path [dashed] (b3) edge (a3); path [dashed] (an) edge (bn);

path (x1) edge (y1); path (y2) edge (x2);

path (y3) edge (x3); path (xn) edge (yn);

end{tikzpicture}

end{center}

end{document} 

Add

draw ($(an)+(0cm,-0.55cm)$) ellipse ({1.5cm} and {4cm});

draw ($(bn)+(0cm,-0.55cm)$) ellipse ({1.5cm} and {4cm});

% begin{large}

    draw[color=white] ($(xn)+(0,-1cm)$) node {color{black}$A$};

    draw[color=white] ($(yn)+(0,-1cm)$) node {color{black}$B$};

%end{large}

to your TikZ drawing code.

You can of course change the numbers in the ellipse-drawing commands if you see that the ellipses are not good enough.

Hope this will help you!