adrianparvino

(a, b) <= (a', b')
  | a /= a' = error "Incomparable"
  | otherwise = b <= b'

adrianparvino

                                                     ,$o$t$$$$$$$$$$,,
                                                  $$T     ,   ,       T$x
                                                $| c.!YT|        T\/ ,   /$,
                                              o|x!                    c ,   $t
                                            o$|                          \    $,
                                           o|                              ,   /t
                                          $|                   $oo$$,      /\   /$,
                                          |                 oT|   ,ocT$,    /|   /$,
                                      o$T TT$c             |   o!|  Toc$     /     $,

adrianparvino

/* This file is part of ASCIIVN.
 *
 * Copyright (C) 2018  Adrian Parvin D. Ouano
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, version 3 of the License.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of

adrianparvino

/* This file is part of ASCIIVN.
 *
 * Copyright (C) 2018  Adrian Parvin D. Ouano
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, version 3 of the License.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of

adrianparvino

struct imagebuffer *
new_imagebuffer(size_t width, size_t height)
{
  size_t size = width * height * color_type_to_bytes(DEFAULT_COLOR_TYPE);
  struct imagebuffer *imagebuffer = malloc(sizeof(struct imagebuffer) + size);

  *imagebuffer = (struct imagebuffer)
    {
      .width = width,
      .height = height,

adrianparvino

Let G be a group and S a subset of G. We shall say that S generates G, or that S is a set of generators for G, if every element of G can be expressed as a product of elements of S or inverses of elements of S, i.e. as a product x_1 \cdots x_n, where each x_i or x_i^-1 is in S. It is clear that the set of all such products is a subgroup of G, and is the smallest subgroup of G containing S. Thus S generates G if and only if the smallest subgroup of G containing S is G itself. If G is generated by S, then we write G = \langle S \rangle

adrianparvino

\documentclass{article}
\usepackage[margin=0in]{geometry}
\usepackage{array}
\setlength\parindent{0pt}
\begin{document}
\begin{tabular}{@{\hspace{0pt}}p{0.5\textwidth}@{\hspace{0pt}}>{\raggedleft}p{0.5\textwidth}@{\hspace{0pt}}}
The quick brown fox & The quick brown fox.
\end{tabular}
\end{document}

adrianparvino

\documentclass{article}
\usepackage[margin=0in]{geometry}
\usepackage{array}
\setlength\parindent{0pt}
\begin{document}
\begin{tabular}{@{\hspace{0pt}}m{0.5\textwidth}@{\hspace{0pt}}>{\raggedleft}m{0.5\textwidth}@{\hspace{0pt}}}
  The quick & {The \\
    quick \\
    brown \\}
\end{tabular}

adrianparvino

A homomorphism whose kernel is trivial is injective. Proof: Given \(f(x) = f(y)\), then \(f(xy^-1) = f(x)f(y^-1) = e’\) Therefore \(xy^-1 = e\) and \(x = y\)

adrianparvino

Notes for page 26

A kernel of f is the set \[\{ x ∈ G | f(x) = e’ \}\] An injective homomorphism \(f:G → G’\) is called an embedding.

A homomorphism whose kernel is trivial is injective.

Proof: