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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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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