% (find-angg "LATEX/2018-2-MD-set-compr.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2018-2-MD-set-compr.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2018-2-MD-set-compr.pdf"))
% (defun e () (interactive) (find-LATEX "2018-2-MD-set-compr.tex"))
% (defun u () (interactive) (find-latex-upload-links "2018-2-MD-set-compr"))
% (find-xpdfpage "~/LATEX/2018-2-MD-set-compr.pdf")
% (find-sh0 "cp -v  ~/LATEX/2018-2-MD-set-compr.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2018-2-MD-set-compr.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2018-2-MD-set-compr.pdf
%               file:///tmp/2018-2-MD-set-compr.pdf
%           file:///tmp/pen/2018-2-MD-set-compr.pdf
% http://angg.twu.net/LATEX/2018-2-MD-set-compr.pdf

% «.mypsection»			(to "mypsection")
% «.picturedots»		(to "picturedots")
% «.comprehension»		(to "comprehension")
% «.comprehension-tables»	(to "comprehension-tables")
% «.comprehension-ex123»	(to "comprehension-ex123")
% «.comprehension-prod»		(to "comprehension-prod")
% «.comprehension-gab»		(to "comprehension-gab")

\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage{color}                % (find-LATEX "edrx15.sty" "colors")
\usepackage{colorweb}             % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
\catcode`\^^J=10                      % (find-es "luatex" "spurious-omega")
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")
\def\expr#1{\directlua{output(tostring(#1))}}
\def\eval#1{\directlua{#1}}
%
\usepackage{edrx15}               % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex            % (find-angg "LATEX/edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-dn4ex "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
\begin{document}

\catcode`\^^J=10

\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
\directlua{dofile "edrxpict.lua"} % (find-LATEX "edrxpict.lua")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end













% «mypsection» (to ".mypsection")
% (find-es "tex" "protect")
% (find-angg ".emacs" "eewrap-mypsection")
% \def\mypsection#1#2{\label{#1}{\bf #2}\ssk}

% (find-es "tex" "page-numbers")
%L psections = {}
%L psectionstex = function ()
%L     local f = function(A)
%L         return format("\\mypsectiontex{%s}{%s}", A[1], A[2])
%L       end
%L     return mapconcat(f, psections, "\n")
%L   end
\def\mypsectiontex#1#2{\par\pageref{#1} #2}
\def\mypsectionstex{\expr{psectionstex()}}
\pu

\def\mypsectionadd#1#2{\directlua{table.insert(psections, {"#1", [[#2]]})}}
\def\mypsection   #1#2{\label{#1}{\bf #2}\mypsectionadd{#1}{#2}\ssk}
%\def\mypsection   #1#2{\label{#1}{\bf #2}\mypsectionadd{#1}{\protect{#2}}\ssk}

% (find-es "tex" "protect")



% «picturedots» (to ".picturedots")
% (find-LATEX "edrxpict.lua" "pictdots")
% (find-LATEX "edrxgac2.tex" "pict2e")
% (to "comprehension-gab")
%
\def\beginpicture(#1,#2)(#3,#4){\expr{beginpicture(v(#1,#2),v(#3,#4))}}
\def\pictaxes{\expr{pictaxes()}}
\def\pictdots#1{\expr{pictdots("#1")}}
\def\picturedots(#1,#2)(#3,#4)#5{%
  \vcenter{\hbox{%
  \beginpicture(#1,#2)(#3,#4)%
  \pictaxes%
  \pictdots{#5}%
  \end{picture}%
  }}%
}

\unitlength=5pt



%   ____                               _                    _
%  / ___|___  _ __ ___  _ __  _ __ ___| |__   ___ _ __  ___(_) ___  _ __
% | |   / _ \| '_ ` _ \| '_ \| '__/ _ \ '_ \ / _ \ '_ \/ __| |/ _ \| '_ \
% | |__| (_) | | | | | | |_) | | |  __/ | | |  __/ | | \__ \ | (_) | | | |
%  \____\___/|_| |_| |_| .__/|_|  \___|_| |_|\___|_| |_|___/_|\___/|_| |_|
%                      |_|
%
% «comprehension» (to ".comprehension")
% (gam181p 5 "comprehension")
\mypsection {comprehension} {``Set comprehensions''}

\def\und#1#2{\underbrace{#1}_{#2}}
\def\und#1#2{\underbrace{#1}_{\text{#2}}}
\def\ug#1{\und{#1}{ger}}
\def\uf#1{\und{#1}{filt}}
\def\ue#1{\und{#1}{expr}}

Notação explícita, com geradores, filtros,

e um ``;'' separando os geradores e filtros da expressão final:

$\begin{array}{lll}
\{\ug{a∈\{1,2,3,4\}}; \ue{10a}\}     &=& \{10,20,30,40\} \\
\{\ug{a∈\{1,2,3,4\}}; \ue{a}\}       &=& \{1,2,3,4\} \\
\{\ug{a∈\{1,2,3,4\}}, \uf{a≥3}; \ue{a}\} &=& \{3,4\} \\
\{\ug{a∈\{1,2,3,4\}}, \uf{a≥3}; \ue{10a}\} &=& \{30,40\} \\
\{\ug{a∈\{10,20\}}, \ug{b∈\{3,4\}}; \ue{a+b}\} &=& \{13,14,23,24\} \\
\{\ug{a∈\{1,2\}}, \ug{b∈\{3,4\}}; \ue{(a,b)}\} &=& \{(1,3),(1,4),(2,3),(2,4)\} \\
\end{array}
$





% (setq last-kbd-macro (kbd "C-w \\ uf{ C-y }"))
% (setq last-kbd-macro (kbd "C-w \\ ue{ C-y }"))

\msk
\msk

Notações convencionais, com ``$|$'' ao invés de ``;'':

Primeiro tipo --- expressão final, ``$|$'', geradores e filtros:

$\begin{array}{lll}
\setofst{10a}{a∈\{1,2,3,4\}} &=&
  \{\ug{a∈\{1,2,3,4\}}; \ue{10a}\} \\
\setofst{10a}{a∈\{1,2,3,4\}, a≥3} &=&
  \{\ug{a∈\{1,2,3,4\}}, \uf{a≥3}; \ue{10a}\} \\
\setofst{a}{a∈\{1,2,3,4\}} &=&
  \{\ug{a∈\{1,2,3,4\}}; \ue{a}\} \\
% \{\ug{a∈\{1,2\}}, \ug{b∈\{3,4\}}; \ue{(a,b)}\} \\
\end{array}
$

\msk

O segundo tipo --- gerador, ``$|$'', filtros ---

pode ser convertido para o primeiro...

o truque é fazer a expressão final ser a variável do gerador:

$\begin{array}{lll}
\setofst{a∈\{1,2,3,4\}}{a≥3} &=& \\
\setofst{a}{a∈\{1,2,3,4\}, a≥3} &=&
  \{\ug{a∈\{1,2,3,4\}}, \uf{a≥3}; \ue{a}\} \\
% \{\ug{a∈\{10,20\}}, \ug{b∈\{3,4\}}; \ue{a+b}\} \\
\end{array}
$

\msk

O que distingue as duas notacões ``$\{\ldots|\ldots\}$'' é

se o que vem antes da ``$|$'' é ou não um gerador.

\bsk

Observações:

$\setofst{\text{gerador}}{\text{filtros}} =
 \{\text{gerador},\text{filtros};\ue{\text{variável do gerador}}\}$

$\setofst{\text{expr}}{\text{geradores e filtros}} =
 \{\text{geradores e filtros}; \text{expr}\}
$

\msk

As notações ``$\{\ldots|\ldots\}$'' são padrão e são usadas em muitos livros de matemática.

A notação ``$\{\ldots;\ldots\}$'' é bem rara; eu aprendi ela em
artigos sobre linguagens de programação, e resolvi apresentar ela aqui
porque acho que ela ajuda a explicar as duas notações
``$\{\ldots|\ldots\}$''.


\newpage

%                                     _                    _               _____
%   ___ ___  _ __ ___  _ __  _ __ ___| |__   ___ _ __  ___(_) ___  _ __   |_   _|
%  / __/ _ \| '_ ` _ \| '_ \| '__/ _ \ '_ \ / _ \ '_ \/ __| |/ _ \| '_ \    | |
% | (_| (_) | | | | | | |_) | | |  __/ | | |  __/ | | \__ \ | (_) | | | |   | |
%  \___\___/|_| |_| |_| .__/|_|  \___|_| |_|\___|_| |_|___/_|\___/|_| |_|   |_|
%                     |_|
%
% «comprehension-tables» (to ".comprehension-tables")
% (gam181p 6 "comprehension-tables")
\mypsection {comprehension-tables} {``Set comprehensions'': como calcular usando tabelas}

\def\tbl#1#2{\fbox{$\begin{array}{#1}#2\end{array}$}}
\def\tbl#1#2{\fbox{$\sm{#2}$}}
\def\V{\mathbf{V}}
\def\F{\mathbf{F}}

% "Stop":
% (find-es "tex" "vrule")
\def\S{\omit$|$\hss}
\def\S{\omit\vrule\hss}
\def\S{\omit\vrule$($\hss}
\def\S{\omit\vrule$\scriptstyle($\hss}
\def\S{\omit\vrule\phantom{$\scriptstyle($}\hss}   % stop

Alguns exemplos:

\msk

\def\s{\mathstrut}
\def\s{\phantom{$|$}}
\def\s{\phantom{|}}
\def\s{}

Se $A := \{x∈\{1,2\}; (x,3-x)\}$

então $A = \{(1,2), (2,1)\}$:

\tbl{ccc}{
 \s x & (x,3-x) \\\hline
 \s 1 & (1,2) \\
 \s 2 & (2,1) \\
}

\msk

Se $I := \{x∈\{1,2,3\}, y∈\{3,4\}, x+y<6; (x,y)\}$

então $I = \{(1,3),(1,4),(1,5)\}$:

\tbl{ccc}{
 \s x & y & x+y<6 & (x,y) \\\hline
 \s 1 & 3 & \V & (1,3) \\
 \s 1 & 4 & \V & (1,4) \\
 \s 2 & 3 & \V & (2,3) \\
 \s 2 & 4 & \F & \S \\
 \s 3 & 3 & \F & \S \\
 \s 3 & 4 & \F & \S \\
}

\msk

Se $D := \setofst{(x,2x)}{x∈\{0,1,2,3\}}$

então $D = \{x∈\{0,1,2,3\}; (x,2x)\}$,

$D = \{(0,0), (1,2), (2,4), (3,6)\}$:

\tbl{ccc}{
 \s x & (x,2x) \\\hline
 \s 0 & (0,0) \\
 \s 1 & (1,2) \\
 \s 2 & (2,4) \\
 \s 3 & (3,6) \\
}

\msk

Se $P := \setofst {(x,y)∈\{1,2,3\}^2} {x≥y}$

então $P = \{(x,y)∈\{1,2,3\}^2, x≥y; (x,y)\}$,

$P = \{(1,1), (2,1), (2,2), (3,1), (3,2), (3,3)\}$:

\tbl{ccc}{
 \s (x,y) & x & y & x≥y & (x,y) \\\hline
 \s (1,1) & 1 & 1 & \V & (1,1) \\
 \s (1,2) & 1 & 2 & \F & \S    \\
 \s (1,3) & 1 & 3 & \F & \S    \\
 \s (2,1) & 2 & 1 & \V & (2,1) \\
 \s (2,2) & 2 & 2 & \V & (2,2) \\
 \s (2,3) & 2 & 3 & \F & \S    \\
 \s (3,1) & 3 & 1 & \V & (3,1) \\
 \s (3,2) & 3 & 2 & \V & (3,2) \\
 \s (3,3) & 3 & 3 & \V & (3,3) \\
}

\bsk

Obs: os exemplos acima correspondem aos

exercícios 2A, 2I, 3D e 5P das próximas páginas.




\newpage

%  _____                   _      _
% | ____|_  _____ _ __ ___(_) ___(_) ___  ___
% |  _| \ \/ / _ \ '__/ __| |/ __| |/ _ \/ __|
% | |___ >  <  __/ | | (__| | (__| | (_) \__ \
% |_____/_/\_\___|_|  \___|_|\___|_|\___/|___/
%
% «comprehension-ex123» (to ".comprehension-ex123")
% (gam181p 7 "comprehension-ex123")
\mypsection {comprehension-ex123} {Exercícios de ``set comprehensions''}

1) Represente graficamente:

$\begin{array}{rcl}
 A & := & \{(1,4), (2,4), (1,3)\} \\
 B & := & \{(1,3), (1,4), (2,4)\} \\
 C & := & \{(1,3), (1,4), (2,4), (2,4)\} \\
 D & := & \{(1,3), (1,4), (2,3), (2,4)\} \\
 E & := & \{(0,3), (1,2), (2,1), (3,0)\} \\
\end{array}
$

\msk

2) Calcule e represente graficamente:

$\begin{array}{rcl}
 A & := & \{x∈\{1,2\}; (x,3-x)\} \\
 B & := & \{x∈\{1,2,3\}; (x,3-x)\} \\
 C & := & \{x∈\{0,1,2,3\}; (x,3-x)\} \\
 D & := & \{x∈\{0,0.5,1, \ldots, 3\}; (x,3-x)\} \\
 E & := & \{x∈\{1,2,3\}, y∈\{3,4\}; (x,y)\} \\
 F & := & \{x∈\{3,4\}, y∈\{1,2,3\}; (x,y)\} \\
 G & := & \{x∈\{3,4\}, y∈\{1,2,3\}; (y,x)\} \\
 H & := & \{x∈\{3,4\}, y∈\{1,2,3\}; (x,2)\} \\
 I & := & \{x∈\{1,2,3\}, y∈\{3,4\}, x+y<6; (x,y)\} \\
 J & := & \{x∈\{1,2,3\}, y∈\{3,4\}, x+y>4; (x,y)\} \\
 K & := & \{x∈\{1,2,3,4\}, y∈\{1,2,3,4\}; (x,y)\} \\
 L & := & \{x,y∈\{0,1,2,3,4\}; (x,y)\} \\
 M & := & \{x,y∈\{0,1,2,3,4\}, y=3; (x,y)\} \\
 N & := & \{x,y∈\{0,1,2,3,4\}, x=2; (x,y)\} \\
 O & := & \{x,y∈\{0,1,2,3,4\}, x+y=3; (x,y)\} \\
 P & := & \{x,y∈\{0,1,2,3,4\}, y=x; (x,y)\} \\
 Q & := & \{x,y∈\{0,1,2,3,4\}, y=x+1; (x,y)\} \\
 R & := & \{x,y∈\{0,1,2,3,4\}, y=2x; (x,y)\} \\
 S & := & \{x,y∈\{0,1,2,3,4\}, y=2x+1; (x,y)\} \\
\end{array}
$

\msk

3) Calcule e represente graficamente:

$\begin{array}{rcl}
 A & := & \setofst{(x,0)}{x∈\{0,1,2,3\}} \\
 B & := & \setofst{(x,x/2)}{x∈\{0,1,2,3\}} \\
 C & := & \setofst{(x,x)}{x∈\{0,1,2,3\}} \\
 D & := & \setofst{(x,2x)}{x∈\{0,1,2,3\}} \\
 E & := & \setofst{(x,1)}{x∈\{0,1,2,3\}} \\
 F & := & \setofst{(x,1+x/2)}{x∈\{0,1,2,3\}} \\
 G & := & \setofst{(x,1+x)}{x∈\{0,1,2,3\}} \\
 H & := & \setofst{(x,1+2x)}{x∈\{0,1,2,3\}} \\
 I & := & \setofst{(x,2)}{x∈\{0,1,2,3\}} \\
 J & := & \setofst{(x,2+x/2)}{x∈\{0,1,2,3\}} \\
 K & := & \setofst{(x,2+x)}{x∈\{0,1,2,3\}} \\
 L & := & \setofst{(x,2+2x)}{x∈\{0,1,2,3\}} \\
 M & := & \setofst{(x,2)}{x∈\{0,1,2,3\}} \\
 N & := & \setofst{(x,2-x/2)}{x∈\{0,1,2,3\}} \\
 O & := & \setofst{(x,2-x)}{x∈\{0,1,2,3\}} \\
 P & := & \setofst{(x,2-2x)}{x∈\{0,1,2,3\}} \\
\end{array}
$


\newpage

%  ____                _                  _
% |  _ \ _ __ ___   __| |   ___ __ _ _ __| |_
% | |_) | '__/ _ \ / _` |  / __/ _` | '__| __|
% |  __/| | | (_) | (_| | | (_| (_| | |  | |_
% |_|   |_|  \___/ \__,_|  \___\__,_|_|   \__|
%
% «comprehension-prod» (to ".comprehension-prod")
% (gam181p 8 "comprehension-prod")
\mypsection {comprehension-prod} {Produto cartesiano de conjuntos}

$A×B:=\{a∈A,b∈B;(a,b)\}$

Exemplo: $\{1,2\}×\{3,4\} = \{(1,3),(1,4),(2,3),(2,4)\}$.

\ssk

Uma notação: $A^2 = A×A$.

Exemplo: $\{3,4\}^2 = \{3,4\}×\{3,4\} = \{(3,3),(3,4),(4,3),(4,4)\}$.

\msk

Sejam:

$A = \{1,2,4\}$,

$B = \{2,3\}$,

$C = \{2,3,4\}$.


\msk

{\bf Exercícios}

\ssk

4) Calcule e represente graficamente:

\begin{tabular}{lll}
a) $A×A$ & d) $B×A$ & g) $C×A$ \\
b) $A×B$ & e) $B×B$ & h) $C×B$ \\
c) $A×C$ & f) $B×C$ & i) $C×C$ \\
\end{tabular}

\msk

5) Calcule e represente graficamente:

$\begin{array}{rcl}
 A &:=& \{x,y∈\{0,1,2,3\};(x,y)\} \\
 B &:=& \{x,y∈\{0,1,2,3\}, y=2; (x,y)\} \\
 C &:=& \{x,y∈\{0,1,2,3\}, x=1; (x,y)\} \\
 D &:=& \{x,y∈\{0,1,2,3\}, y=x; (x,y)\} \\
 E &:=& \{x,y∈\{0,1,2,3,4\}, y=2x; (x,y)\} \\
 F &:=& \{(x,y)∈\{0,1,2,3,4\}^2, y=2x; (x,y)\} \\
 G &:=& \{(x,y)∈\{0,1,2,3,4\}^2, y=x; (x,y)\} \\
 H &:=& \{(x,y)∈\{0,1,2,3,4\}^2, y=x/2; (x,y)\} \\
 I &:=& \{(x,y)∈\{0,1,2,3,4\}^2, y=x/2+1; (x,y)\} \\
 J &:=& \setofst {(x,y)∈\{0,1,2,3,4\}^2} {y=2x} \\
 K &:=& \setofst {(x,y)∈\{0,1,2,3,4\}^2} {y=x} \\
 L &:=& \setofst {(x,y)∈\{0,1,2,3,4\}^2} {y=x/2} \\
 M &:=& \setofst {(x,y)∈\{0,1,2,3,4\}^2} {y=x/2+1} \\
 N &:=& \setofst {(x,y)∈\{1,2,3\}^2} {0x+0y=0} \\
 O &:=& \setofst {(x,y)∈\{1,2,3\}^2} {0x+0y=2} \\
 P &:=& \setofst {(x,y)∈\{1,2,3\}^2} {x≥y} \\
 \end{array}
$

\msk

6) Represente graficamente:

$\begin{array}{rcl}
 J' &:=& \setofst {(x,y)∈\R^2} {y=2x} \\
 K' &:=& \setofst {(x,y)∈\R^2} {y=x} \\
 L' &:=& \setofst {(x,y)∈\R^2} {y=x/2} \\
 M' &:=& \setofst {(x,y)∈\R^2} {y=x/2+1} \\
 N' &:=& \setofst {(x,y)∈\R^2} {0x+0y=0} \\
 O' &:=& \setofst {(x,y)∈\R^2} {0x+0y=2} \\
 P' &:=& \setofst {(x,y)∈\R^2} {x≥y} \\
 \end{array}
$





\newpage

%   ____       _                _ _
%  / ___| __ _| |__   __ _ _ __(_) |_ ___
% | |  _ / _` | '_ \ / _` | '__| | __/ _ \
% | |_| | (_| | |_) | (_| | |  | | || (_) |
%  \____|\__,_|_.__/ \__,_|_|  |_|\__\___/
%
% «comprehension-gab» (to ".comprehension-gab")
% (gam181p 9 "comprehension-gab")
% (to "picturedots")
\mypsection {comprehension-gab} {Gabarito dos exercícios de set comprehensions}

% \bhbox{$\picturedots(-1,-2)(5,5){ 3,1 3,2 3,3 }$}

1)
$
A = B = C = \picturedots(0,0)(3,4){ 1,4 2,4 1,3 }
\quad
D = \picturedots(0,0)(3,4){ 1,4 2,4 1,3 2,3 }
\quad
E = \picturedots(0,0)(4,4){ 0,3 1,2 2,1 3,0 }
$

\bsk

2)
$     A = \picturedots(0,0)(4,4){     1,2 2,1     }
\quad B = \picturedots(0,0)(4,4){     1,2 2,1 3,0 }
\quad C = \picturedots(0,0)(4,4){ 0,3 1,2 2,1 3,0 }
\quad D = \picturedots(0,0)(4,4){ 0,3 .5,2.5 1,2 1.5,1.5 2,1 2.5,.5 3,0 }
$

\msk

$
\quad E = \picturedots(0,0)(4,4){ 1,3 2,3 3,3   1,4 2,4 3,4 }
\quad F = \picturedots(0,0)(4,4){ 3,1 4,1   3,2 4,2   3,3 4,3 }
\quad G = \picturedots(0,0)(4,4){ 1,3 2,3 3,3   1,4 2,4 3,4 }
\quad H = \picturedots(0,0)(4,4){ 3,2 4,2 }
\quad I = \picturedots(0,0)(4,4){ 1,3 2,3       1,4         }
\quad J = \picturedots(0,0)(4,4){     2,3 3,3   1,4 2,4 3,4 }
$

\msk

$
\quad K = \picturedots(0,0)(4,4){     1,4 2,4 3,4 4,4
                                      1,3 2,3 3,3 4,3
                                      1,2 2,2 3,2 4,2
                                      1,1 2,1 3,1 4,1 }
\quad L = \picturedots(0,0)(4,4){ 0,4 1,4 2,4 3,4 4,4
                                  0,3 1,3 2,3 3,3 4,3
                                  0,2 1,2 2,2 3,2 4,2
                                  0,1 1,1 2,1 3,1 4,1
                                  0,0 1,0 2,0 3,0 4,0 }
\quad M = \picturedots(0,0)(4,4){ 0,3 1,3 2,3 3,3 4,3 }
\quad N = \picturedots(0,0)(4,4){ 2,0 2,1 2,2 2,3 2,4 }
\quad O = \picturedots(0,0)(4,4){ 0,3 1,2 2,1 3,0 }
\quad P = \picturedots(0,0)(4,4){ 0,0 1,1 2,2 3,3 4,4 }
$

\msk

$
\quad Q = \picturedots(0,0)(4,4){ 0,1 1,2 2,3 3,4 }
\quad R = \picturedots(0,0)(4,4){ 0,0 1,2 2,4 }
\quad S = \picturedots(0,0)(4,4){ 0,1 1,3 }
$

\bsk

3)
$     A = \picturedots(0,0)(4,4){ 0,0 1,0  2,0 3,0   }
\quad B = \picturedots(0,0)(4,4){ 0,0 1,.5 2,1 3,1.5 }
\quad C = \picturedots(0,0)(4,4){ 0,0 1,1  2,2 3,3   }
\quad D = \picturedots(0,0)(4,7){ 0,0 1,2  2,4 3,6   }
$

$
\quad E = \picturedots(0,0)(4,4){ 0,1 1,1   2,1 3,1   }
\quad F = \picturedots(0,0)(4,4){ 0,1 1,1.5 2,2 3,2.5 }
\quad G = \picturedots(0,0)(4,4){ 0,1 1,2   2,3 3,4   }
\quad H = \picturedots(0,0)(4,7){ 0,1 1,3   2,5 3,7   }
$

$
\quad I = \picturedots(0,0)(4,4){ 0,2 1,2   2,2 3,2   }
\quad J = \picturedots(0,0)(4,4){ 0,2 1,2.5 2,3 3,3.5 }
\quad K = \picturedots(0,0)(4,4){ 0,2 1,3   2,4 3,5   }
\quad L = \picturedots(0,0)(4,8){ 0,2 1,4   2,6 3,8   }
$

$
\quad M = \picturedots(0,0)(4,4){ 0,2 1,2   2,2 3,2   }
\quad N = \picturedots(0,0)(4,4){ 0,2 1,1.5 2,1 3,.5 }
\quad O = \picturedots(0,-1)(4,4){ 0,2 1,1   2,0 3,-1  }
\quad P = \picturedots(0,-5)(4,3){ 0,2 1,0   2,-2 3,-4   }
$

\bsk

4)
$     A×A = \picturedots(0,0)(4,4){ 1,1 2,1 4,1   1,2 2,2 4,2   1,4 2,4 4,4 }
\quad B×A = \picturedots(0,0)(4,4){ 2,1 3,1       2,2 3,2       2,4 3,4     }
\quad C×A = \picturedots(0,0)(4,4){ 2,1 3,1 4,1   2,2 3,2 4,2   2,4 3,4 4,4 }
$

\msk

$
\quad A×B = \picturedots(0,0)(4,4){ 1,2 2,2 4,2   1,3 2,3 4,3 }
\quad B×B = \picturedots(0,0)(4,4){ 2,2 3,2       2,3 3,3     }
\quad C×B = \picturedots(0,0)(4,4){ 2,2 3,2 4,2   2,3 3,3 4,3 }
$

\msk

$
\quad A×C = \picturedots(0,0)(4,4){ 1,2 2,2 4,2   1,3 2,3 4,3   1,4 2,4 4,4 }
\quad B×C = \picturedots(0,0)(4,4){ 2,2 3,2       2,3 3,3       2,4 3,4     }
\quad C×C = \picturedots(0,0)(4,4){ 2,2 3,2 4,2   2,3 3,3 4,3   2,4 3,4 4,4 }
$

\bsk

5)
$     A = \picturedots(0,0)(4,4){ 0,3 1,3 2,3 3,3
                                  0,2 1,2 2,2 3,2
                                  0,1 1,1 2,1 3,1
                                  0,0 1,0 2,0 3,0 }
\quad B = \picturedots(0,0)(4,4){ 0,2 1,2 2,2 3,2 }
\quad C = \picturedots(0,0)(4,4){ 1,0 1,1 1,2 1,3 }
\quad D = \picturedots(0,0)(4,4){ 0,0 1,1 2,2 3,3 }
\quad E = \picturedots(0,0)(4,4){ 0,0 1,2 2,4 }
$

\msk

$
\quad F = \picturedots(0,0)(4,4){ 0,0 1,2 2,4         }
\quad G = \picturedots(0,0)(4,4){ 0,0 1,1 2,2 3,3 4,4 }
\quad H = \picturedots(0,0)(4,4){ 0,0 2,1 4,2         }
\quad I = \picturedots(0,0)(4,4){ 0,1 2,2 4,3         }
$

\msk

$
\quad J = \picturedots(0,0)(4,4){ 0,0 1,2 2,4         }
\quad K = \picturedots(0,0)(4,4){ 0,0 1,1 2,2 3,3 4,4 }
\quad L = \picturedots(0,0)(4,4){ 0,0 2,1 4,2         }
\quad M = \picturedots(0,0)(4,4){ 0,1 2,2 4,3         }
$

\msk

$
\quad N = \picturedots(0,0)(4,4){ 1,3 2,3 3,3
                                  1,2 2,2 3,2
                                  1,1 2,1 3,1 }
\quad O = \picturedots(0,0)(4,4){             }
\quad P = \picturedots(0,0)(4,4){         3,3
                                      2,2 3,2
                                  1,1 2,1 3,1 }
$




\end{document}

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