% (find-angg "LATEX/2018-2-MD-P2B.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2018-2-MD-P2B.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2018-2-MD-P2B.pdf"))
% (defun e () (interactive) (find-LATEX "2018-2-MD-P2B.tex"))
% (defun u () (interactive) (find-latex-upload-links "2018-2-MD-P2B"))
% (find-xpdfpage "~/LATEX/2018-2-MD-P2B.pdf")
% (find-sh0 "cp -v  ~/LATEX/2018-2-MD-P2B.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2018-2-MD-P2B.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2018-2-MD-P2B.pdf
%               file:///tmp/2018-2-MD-P2B.pdf
%           file:///tmp/pen/2018-2-MD-P2B.pdf
% http://angg.twu.net/LATEX/2018-2-MD-P2B.pdf
\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{dednat6dir = "dednat6/"}
%\directlua{dofile(dednat6dir.."dednat6.lua")}
%\directlua{texfile(tex.jobname)}
%\directlua{verbose()}
%\directlua{output(preamble1)}
%\def\expr#1{\directlua{output(tostring(#1))}}
%\def\eval#1{\directlua{#1}}
%\def\pu{\directlua{pu()}}

\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



\def\V{\mathbf{V}}
\def\F{\mathbf{F}}

\def\Par  {\mathsf{par}}
\def\Impar{\mathsf{impar}}





%   ____      _                    _ _           
%  / ___|__ _| |__   ___  ___ __ _| | |__   ___  
% | |   / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \ 
% | |__| (_| | |_) |  __/ (_| (_| | | | | | (_) |
%  \____\__,_|_.__/ \___|\___\__,_|_|_| |_|\___/ 
%                                                

{\setlength{\parindent}{0em}
\footnotesize
\par Matemática Discreta
\par PURO-UFF - 2018.2
\par P2 - 12/dez/2018 - Eduardo Ochs
\par Turma pequena (C1, com aulas nas terças e quartas)
\par Proibido usar quaisquer aparelhos eletrônicos.

}

\bsk
\bsk

\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\B       (#1 pts){{\bf(#1 pts)}}
% Usage:
% 1) \T(Total: 2.34 pts) Foo
% a) \B(0.45 pts) Bar




{
\setlength{\parindent}{0em}


1) \T(Total: 2.0 pts) Mostre que uma das regras abaixo é admissível e
a outra não.
%:
%:
%:   Γ⊢A→B   Γ⊢¬B       Γ,A⊢C
%:   ------------(R1)   -------(R2)
%:       Γ⊢¬A           Γ,A∨B⊢C
%:
%:       ^R1              ^R2
%:
\pu
$$\ded{R1} \qquad \ded{R2}$$


\bsk




2) \T(Total: 3.5 pts) Para cada $k∈\N$ Seja $P(k)$ a seguinte
proposição: $1+3+\ldots+(2k+1)=(k+1)^2$.

a) \B(0.5 pts) Defina uma função $f(k)$ sem `$\ldots$'s, usando
somatório, tal que $f(k)=1+3+\ldots+(2n+1)$.

b) \B(0.5 pts) Calcule $f(21)-f(20)$.

c) \B(0.5 pts) Calcule $f(0)$ e $(f(0)=0^2)$.

d) \B(1.5 pts) Demonstre $k∈N⊢P(k)→P(k+1)$.

e) \B(0.5 pts) Demonstre $∀n∈N.P(n)$.


\bsk
\bsk


3) \T(Total: 2.0 pts) Seja $F:\N×\N→\N$ uma função que obedece:

(F0) $F(0,0)=0$,

(FT) $∀n∈\N.0<n→F(n+1,0)=F(0,n)+1$,

(FS) $∀x,y∈\N.0<y<x→F(x,y)=F(x,y-1)+1$,

(FE) $∀x,y∈\N.0≤x≤y→F(x,y)=F(x+1,y)+1$,

Calcule os valores de $F(x,y)$ para todos os $(x,y)∈\{0,1,2,3\}^2$.


\bsk
\bsk


4) \T(Total: 2.5 pts) Seja $A={1,2,3,4,5,6}$ e $f,g:A→A$ as seguintes
permutações:

$f = (1\; 2\; 3)(4\; 5)$ \qquad $(=\{(1,2), (2,3), (3,1), (4,5), (5,4), (6,6)\})$

$g = (3\; 4)$

a) \B(0.5 pts) Calcule $f∘g$ e $g∘f$ e expresse-os na notação de ciclos.

b) \B(1.0 pts) Calcule $f, f^2, f^3, f^4, f^5, f^6$ e expresse-os na notação de ciclos.

c) \B(1.0 pts) Calcule $f^{20}$ e expresse-o na notação de ciclos.


}




\end{document}

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