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// Copyright (c) 2017-2020 NVIDIA Corporation
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_NV_geometry_shader_passthrough.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2017-02-15
*Interactions and External Dependencies*::
- This extension provides API support fo... | vulkan-spec/appendices/VK_NV_geometry_shader_passthrough.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.183 | false | false | 5,357 | -1 |
// Copyright 2024-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_NV_raw_access_chains.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2023-12-04
*Interactions and External Dependencies*::
- This extension requires
{spirv}/NV/SPV_NV_ra... | vulkan-spec/appendices/VK_NV_raw_access_chains.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.375 | false | false | 966 | -1 |
// Copyright 2014-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_EXT_directfb_surface.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2020-06-16
*IP Status*::
No known IP claims.
*Contributors*::
- Nicolas Caramelli
=== Description
... | vulkan-spec/appendices/VK_EXT_directfb_surface.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.381 | false | false | 770 | -1 |
// Copyright 2023-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_ANDROID_external_format_resolve.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2023-05-03
*IP Status*::
No known IP claims.
*Contributors*::
- Tobias Hector, AMD
- Ch... | vulkan-spec/appendices/VK_ANDROID_external_format_resolve.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.267 | false | false | 967 | -1 |
// Copyright (c) 2020-2023 Huawei Technologies Co. Ltd.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_HUAWEI_cluster_culling_shader.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2023-08-16
*Interactions and External Dependencies*::
- This extension provides API s... | vulkan-spec/appendices/VK_HUAWEI_cluster_culling_shader.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.248 | false | false | 8,163 | -1 |
// Copyright 2015-2026 The Khronos Group Inc.
// SPDX-License-Identifier: CC-BY-4.0
// This text fragment is used many times in the Features chapter.
ifdef::hidden[]
// tag::features[]
If the sname:{refpage} structure is included in the pname:pNext chain of the
slink:VkPhysicalDeviceFeatures2 structure passed to
flink... | vulkan-spec/chapters/features.adoc#p0 | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.214 | false | false | 149,969 | -1 |
// Copyright 2021 NVIDIA Corporation
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_EXT_application_parameters.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2021-12-14
*Contributors*::
- Daniel Koch, NVIDIA
- Jonathan Mccaffrey, NVIDIA
- Aidan Fabius, Lynx
==... | vulkan-spec/appendices/VK_EXT_application_parameters.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.305 | false | false | 2,452 | -1 |
// Copyright 2019-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
// Common Valid Usage
// Common to acceleration structure build commands
* [[VUID-{refpage}-mode-04628]]
The pname:mode member of each element of pname:pInfos must: be a valid
elink:VkBuildAccelerationStructureModeKHR valu... | vulkan-spec/chapters/commonvalidity/build_acceleration_structure_common.adoc | books | vulkan-spec | vulkan-spec/chapters/commonvalidity | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.166 | false | false | 17,165 | -1 |
// Copyright 2017-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_KHR_vulkan_memory_model.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2018-12-10
*IP Status*::
No known IP claims.
*Contributors*::
- Jeff Bolz, NVIDIA
- Alan Baker,... | vulkan-spec/appendices/VK_KHR_vulkan_memory_model.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.216 | false | false | 2,008 | -1 |
// Copyright 2023-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_EXT_external_memory_acquire_unmodified.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2023-03-09
*Contributors*::
- Lina Versace, Google
- Chia-I Wu, Google
- James Jon... | vulkan-spec/appendices/VK_EXT_external_memory_acquire_unmodified.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.304 | false | false | 926 | -1 |
// Copyright 2015-2026 The Khronos Group Inc.
// SPDX-License-Identifier: CC-BY-4.0
[[primsrast]]
= Rasterization
Rasterization is the process by which a primitive is converted to a
two-dimensional image.
Each discrete location of this image contains associated data such as depth,
color, or other attributes.
Rasteri... | vulkan-spec/chapters/primsrast.adoc#p0 | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.144 | false | false | 149,481 | -1 |
// Copyright 2015-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_EXT_debug_report.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2020-12-14
*IP Status*::
No known IP claims.
*Contributors*::
- Courtney Goeltzenleuchter, LunarG
- Da... | vulkan-spec/appendices/VK_EXT_debug_report.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.253 | false | false | 7,641 | -1 |
// Copyright (c) 2023 Arm Ltd
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_ARM_render_pass_striped.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2023-11-21
*IP Status*::
No known IP claims.
*Contributors*::
- Jan-Harald Fredriksen, Arm
- Lisa Wu, Arm
- T... | vulkan-spec/appendices/VK_ARM_render_pass_striped.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.32 | false | false | 721 | -1 |
// Copyright 2023-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
// Common Valid Usage
// common to all pipeline creation functions
ifndef::VKSC_VERSION_1_0[]
ifdef::VK_KHR_pipeline_binary[]
* [[VUID-{refpage}-pNext-09616]]
If slink:VkPipelineBinaryInfoKHR::pname:binaryCount is not `0` for ... | vulkan-spec/chapters/commonvalidity/pipeline_create_common.adoc | books | vulkan-spec | vulkan-spec/chapters/commonvalidity | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.245 | false | false | 2,668 | -1 |
// Copyright (c) 2018-2020 NVIDIA Corporation
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_NV_cuda_kernel_launch.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2020-09-30
*Contributors*::
- Eric Werness, NVIDIA
=== Description
Interoperability between APIs can ... | vulkan-spec/appendices/VK_NV_cuda_kernel_launch.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.137 | false | false | 2,560 | -1 |
// Copyright 2017-2026 The Khronos Group Inc.
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_EXT_present_timing.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2024-10-09
*IP Status*::
No known IP claims.
*Contributors*::
- Ian Elliott, Google
- James Jones, NVID... | vulkan-spec/appendices/VK_EXT_present_timing.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.105 | false | false | 9,326 | -1 |
// Copyright 2014-2026 The Khronos Group Inc.
//
// SPDX-License-Identifier: CC-BY-4.0
include::{generated}/meta/{refprefix}VK_KHR_shader_float_controls.adoc[]
=== Other Extension Metadata
*Last Modified Date*::
2018-09-11
*IP Status*::
No known IP claims.
*Contributors*::
- Alexander Galazin, Arm
- Jan-... | vulkan-spec/appendices/VK_KHR_shader_float_controls.adoc | books | vulkan-spec | vulkan-spec | CC-BY-4.0 | CC-BY-4.0 | b94f8bd2d004d2949469d8a5071ee10eb7d23566 | doc | false | 0.27 | false | false | 4,930 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_addtops-genfun">
<title>Generating Functions</title>
<idx><h>generating function</h></idx>
<introduction>
<p>
There is an extremely powerful tool in discrete mathematics used to manipulate sequences c... | discrete-book/source/sec_addtops-genfun.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 26,102 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-conc">
<title>Chapter Review</title>
<exercise>
<statement>
<p>
Which (if any) of the graphs below are the same?
Which are different?
Explain.
</p>
<sideby... | discrete-book/source/exercises/gt-conc.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 32,087 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_seq_basics">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Consider the sequence <m>(a_n)_{n \ge 1}</m> that starts <m>1, 3, 5, 7, 9, \ldots</m> (i.e., the odd numb... | discrete-book/source/exercises/seq-basics.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 15,102 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_seq-polynomial">
<title>Polynomial Sequences</title>
<idx><h>polynomial fitting</h></idx>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following... | discrete-book/source/sec_seq-polynomial.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 37,465 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_structures-functions">
<title>Functions</title>
<introduction>
<p>
This section contains many details about functions, much of which should be familiar from non-discrete contexts.
As you read, co... | discrete-book/source/sec_structures-functions.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.005 | true | false | 49,386 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_trees">
<title>Trees</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
Prove basic... | discrete-book/source/sec_gt-trees.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 36,833 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<frontmatter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="frontmatter">
<titlepage>
<author>
<personname>
Oscar Levin</personname>
<department>Mathematical Sciences</department>
<institution>University of Northern Colorado</institution>
</a... | discrete-book/source/frontmatter.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.01 | true | false | 13,015 | -1 |
\def\circleA{(-.5,0) circle (1)}
\def\circleAlabel{(-1.5,.6) node[above]{$A$}}
\def\circleB{(.5,0) circle (1)}
\def\circleBlabel{(1.5,.6) node[above]{$B$}}
\def\circleC{(0,-1) circle (1)}
\def\circleClabel{(.5,-2) node[right]{$C$}}
\def\twosetbox{(-2,-1.4) rectangle (2,1.4)}
\def\threesetbox{(-2.5,-2.4) rectangle (2.5,... | discrete-book/source/assets/tikz-defs.tex | math | discrete-book | discrete-book/source/assets | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.048 | true | false | 96,523 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_seq_intro">
<title>Describing Sequences</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
... | discrete-book/source/sec_seq-basics.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.005 | true | false | 42,315 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_matchings">
<title>Matching in Bipartite Graphs</title>
<idx>
<h>matching</h>
</idx>
<idx>
<h>graph</h>
<h>matching</h>
</idx>
<idx>
<h>perfect matching</h>
<see>matching</see>
</idx... | discrete-book/source/sec_gt-matchings.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 10,672 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_logic-rules">
<title>Rules of Logic</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
Us... | discrete-book/source/sec_logic-rules.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 48,813 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_graphtheory">
<title>Graph Theory</title>
<introduction>
<p>
Graph theory has existed as a branch of mathematics for only a short time; the first book on graph theory was published less than 100 years ag... | discrete-book/source/ch_graphtheory.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 1,339 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_intro-sets">
<exercise label="ww-sets-ops">
<webwork source="Contrib/DMOI/0-Introduction_and_Preliminaries/0_3-Sets/0_3_1.pg"/>
</exercise>
<exercise label="ww-sets-least">
<webwork>
<pg-... | discrete-book/source/exercises/intro-sets.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.013 | true | false | 25,855 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-trees">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Which of the following graphs are trees?
<ol>
<li>
<p>
<m>G = (V, E)</m> with <m>V = ... | discrete-book/source/exercises/gt-trees.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.002 | true | false | 14,888 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-matchings">
<exercise>
<statement>
<p>
Find a matching of the bipartite graphs below or explain why no matching exists.
</p>
<sidebyside widths="20% 30% 40%" margins="auto"... | discrete-book/source/exercises/gt-matchings.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 16,240 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_comb-proofs">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Give a combinatorial proof of the identity <m>2+2+2 = 3\cdot 2</m>.
</p>
</statement>
<solution>... | discrete-book/source/exercises/counting-proofs.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 18,661 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_logic-implications">
<title>Implications</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
... | discrete-book/source/sec_logic-implications.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 38,891 | -1 |
\thispagestyle{empty}
\vskip 2em
\noindent Oscar Levin \\ Mathematical Science \\ University of Northern Colorado \\ Greeley, Co 80639 \\ \nolinkurl{oscar.levin@unco.edu} \\ \url{http://math.oscarlevin.com/}
\vfill
\vfill
\noindent\textcopyright ~ 2013-2025 by Oscar Levin
\vskip 3em
\noindent
This work is licensed... | discrete-book/assets/frontmatter/copyright-page.tex | math | discrete-book | discrete-book/assets/frontmatter | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.217 | true | false | 904 | -1 |
# Errata for 3rd Edition
Many errors were fixed in preparation of the 3rd edition, but thanks in part to the careful reading by students and instructors, more have been revealed. I'm releasing occasional fixes to the html, pdf and print versions. These updates should not change the pagination in any major way.
If ... | discrete-book/errata3.md | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.775 | true | false | 5,289 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_counting-non-disjoint">
<title>Non-Disjoint Outcomes</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li... | discrete-book/source/sec_counting-non-disjoint.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 35,484 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises-logic-rules">
<title>Additional Exercises</title>
<exercise>
<idx><h>knights and knaves</h>
</idx>
<idx><h>puzzle</h><h>knights and knaves</h>
</idx>
<statement>
<p>
... | discrete-book/source/exercises/logic-rules.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 17,372 | -1 |
# Errata for 2nd Edition
Many errors were fixed in preparation of the 2nd edition, but thanks in part to the careful reading by students and instructors, more have been revealed. Over Summer 2017, I corrected many of these errors and released a corrected 2nd edition in time for Fall 2017. The corrected edition is m... | discrete-book/errata2.md | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.821 | true | false | 2,897 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-paths">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
You and your friends want to tour the southwest by car.
You will visit the nine states below,
wi... | discrete-book/source/exercises/gt-paths.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 15,172 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_counting-multisets">
<title>Counting Multisets</title>
<idx> <h>sticks and stones</h> </idx>
<idx><h>multisets (counting)</h></idx>
<idx><h>balls and bins</h><see>sticks and stones</see></idx>
<idx><h>star... | discrete-book/source/sec_counting-multisets.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 30,291 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-planar">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Is it possible for a planar graph to have 6 vertices, 10 edges, and 5 faces?
Explain.
</p>
... | discrete-book/source/exercises/gt-planar.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 11,384 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_seq-conc">
<title>Chapter Review</title>
<exercise label="ww-seq-sum">
<webwork source="Contrib/DMOI/2-Sequences/2_6-Chapter_Summary/2_6_1.pg" />
</exercise>
<exercise label="ww-seq-arith">
<w... | discrete-book/source/exercises/seq-conc.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 14,250 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_sequences">
<title>Sequences</title>
<introduction>
<p>
We have encountered <em>finite</em> sequences already as a discrete structure and something we can count. In this chapter, we will consider possib... | discrete-book/source/ch_sequences.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 4,525 | -1 |
# Discrete Mathematics: An Open Introduction
An open textbook for Discrete Mathematics, as taught at the University of Northern Colorado. More information about the project is available on [the book's main website](http://discrete.openmathbooks.org).
The `main` branch now has the source code for the 3rd edition. Work... | discrete-book/README.md | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.162 | true | false | 2,554 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_counting-combine-outcomes">
<title>Combining Outcomes</title>
<idx><h>sum principle</h></idx>
<idx><h>product principle</h></idx>
<idx><h>additive principle</h><see>sum principle</see></idx>
<idx><h>multiplicativ... | discrete-book/source/sec_counting-combine-outcomes.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 41,648 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_seq-strong-induction">
<title>Additional Exercises</title>
<exercise>
<idx><h>football (American)</h></idx>
<statement>
<p>
Suppose a football team only scores 3-point field goals... | discrete-book/source/exercises/seq-strong-induction.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 8,498 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_logic-statements">
<title>Mathematical Statements</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
... | discrete-book/source/sec_logic-statements.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 50,907 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_advPIE">
<title>Advanced Counting Using PIE</title>
<idx>principle of inclusion/exclusion</idx>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
... | discrete-book/source/sec_counting-advPIE.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 38,933 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_seq-growth">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Suppose that the candy machine currently holds exactly 650 Skittles,
and every time someone in... | discrete-book/source/exercises/seq-growth.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 7,561 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_sequences-conc">
<title>Chapter Summary</title>
<introduction>
<investigation>
<p>
Each day your supply of magic chocolate-covered espresso beans doubles
(each one splits in half),
... | discrete-book/source/sec_seq-conc.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 4,783 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-intro">
<title>Additional Exercises</title>
<exercise>
<idx><h>handshakes</h><h>connection to graphs</h> </idx>
<statement>
<p>
If 10 people each shake hands with each other, how many handsh... | discrete-book/source/exercises/gt-intro.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 29,193 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xml:id="sec_gt-relations" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Relations and Graphs</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
... | discrete-book/source/sec_gt-relations.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 49,553 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_addtops-numbth">
<exercise>
<statement>
<p>
Suppose <m>a</m>, <m>b</m>, and <m>c</m> are integers.
Prove that if <m>a \mid b</m>, then <m>a \mid bc</m>.
</p>
</statement>... | discrete-book/source/exercises/addtops-numbth.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 8,416 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_solutions">
<title>Solutions to Exercises</title>
<p>
IIII
</p>
<p>
\printnoidxglossary[type=symbols,style=myglosses,
nogroupskip=true, sort=def]
</p>
<p>
\listoftodos
</p>
</chapter>
| discrete-book/source/ch_solutions.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 320 | -1 |
# Todo list
## 0.3:
* Randomize 23.
## 0.4:
* Randomize 1.
## 2.1:
* Problem 2 can be made into ww easily.
* Problem 3 as well (maybe not as easy).
* And 4. Webwork can do sequences. Look at 2.2.2 as example.
* None of these appear to be randomized/different. Do some!
## 2.2:
- Problem 1 should be ww (even if n... | discrete-book/webwork/todo.md | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.667 | true | false | 611 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_counting-pascal">
<title>Pascal's Arithmetical Triangle</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li... | discrete-book/source/sec_counting-pascal.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.007 | true | false | 43,142 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xml:id="ex-counting-probability" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
When playing 5-card poker, a <term>full house</term> is a hand that contains three cards of one rank and t... | discrete-book/source/exercises/counting-probability.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 4,068 | -1 |
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\usepackage{textcomp}
\usepackage{multicol}
\tcbset{ divisionsolutionstyle/.style={bwminimalstyle, runintitlest... | discrete-book/assets/latex-preamble-styles-crc.tex | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.923 | true | false | 8,679 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_gt-paths">
<title>Euler Trails and Circuits</title>
<idx><h>trail</h><h>Euler</h></idx>
<idx><h>path</h><h>Hamilton</h></idx>
<idx><h>circuit</h><h>Euler</h></idx>
<idx><h>cycle</h><h>Hamilton</h></idx>
<... | discrete-book/source/sec_gt-paths.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 26,559 | -1 |
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<ta... | discrete-book/project.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 2,096 | -1 |
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<exercises xml:id="ex-logic-implications" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Additional Exercises</title>
<exercise>
<statement>
<p> Translate into English: <ol>
<li><m>\forall x (E(x) \imp E(x +2))</m>.</li>
<li><m>\forall x \... | discrete-book/source/exercises/logic-implications.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 7,506 | -1 |
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<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_structures-sets">
<title>Sets</title>
<introduction>
<p>
Note that this section contains a lot of detail on what we can say about sets. If you have never seen anything about unions or Venn diagrams, it... | discrete-book/source/sec_structures-sets.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 42,971 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<chapter xml:id="ch_structures" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Discrete Structures Revisited</title>
<introduction>
<p>
Throughout the previous chapters, we have seen many examples of discrete structures and their properties and uses. The follo... | discrete-book/source/ch_structures.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 731 | -1 |
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<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_logic-conc">
<title>Chapter Summary</title>
<introduction>
<p>
We have considered logic both as its own sub-discipline of mathematics
and as a means to help us better understand and write proofs.
... | discrete-book/source/sec_logic-conc.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 4,094 | -1 |
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<statement>
<p> For a given predicate <m>P(x)</m>, you might believe that the statements <m>\forall x P(x)</m> or <m>\exist... | discrete-book/source/exercises/logic-proofs.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 22,476 | -1 |
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<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_counting-binom">
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<exercise>
<p>
How many lattice paths are there from <m>(0,0)</m> to <m>(8,3)</m>. How many lattice paths are there from <m>(0,0)</m> to <m>(3,8)... | discrete-book/source/exercises/counting-pascal.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 2,846 | -1 |
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<introduction>
<investigation>
<p>
Suppose you have a huge box of animal crackers containing plenty of each of 10 different animals.
For the c... | discrete-book/source/sec_counting-conc.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 5,929 | -1 |
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<exercise>
<statement>
<p>
Prove that for any two sets <m>A</m> and <m>B</m>, <m>A \subseteq B</m> if and only if <m>A \cup B = B<... | discrete-book/source/exercises/logic-structures.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 7,221 | -1 |
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<section xml:id="sec-counting-probability" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Applications to Probability</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
... | discrete-book/source/sec_counting-probability.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.004 | true | false | 45,938 | -1 |
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<idx><h>recurrence relation</h><h>solving</h></idx>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the fo... | discrete-book/source/sec_seq-exponential.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 26,724 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_gt-planar">
<title>Planar Graphs</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
... | discrete-book/source/sec_gt-planar.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 36,048 | -1 |
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<exercise>
<statement>
<p>
Let <m>A</m>, <m>B</m>, and <m>C</m> be sets.
<ol>
<li>
Find <m>\card{(A \... | discrete-book/source/exercises/counting-non-disjoint.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 1,665 | -1 |
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<statement>
<p>
Based on the previous question,
give a combinatorial proof for the identity:
<me>
... | discrete-book/source/exercises/counting-advPIE.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 1,963 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_counting-combperm">
<title>Combinations and Permutations</title>
<idx><h>combination</h>
</idx>
<idx><h>permutation</h>
</idx>
<objectives>
<introduction>
<p>
After completing this section, you should b... | discrete-book/source/sec_counting-combperm.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 36,063 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_intro-intro">
<title>What is Discrete Mathematics?</title>
<blockquote>
<p>
dis<midpoint/>crete / dis'krët.
</p>
<p>
<em>Adjective</em>: Individually separate and distinct.
</p>
<p>
<em>Synonyms... | discrete-book/source/sec_intro-intro.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 9,912 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_seq-growth">
<title>Rate of Growth</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
... | discrete-book/source/sec_seq-growth.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 42,795 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_intro-functions">
<exercise label="ww-functions-two-line">
<webwork source="Contrib/DMOI/0-Introduction_and_Preliminaries/0_4-Functions/0_4_1.pg" />
</exercise>
<exercise label="ww-functions-inj-su... | discrete-book/source/exercises/intro-functions.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.004 | true | false | 29,413 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_seq-induction">
<title>Proof by Induction</title>
<idx>
<h>proof</h>
<h>by induction</h>
</idx>
<idx>
<h>induction</h>
</idx>
<idx>
<h>mathematical induction</h>
<see>induction</see>
... | discrete-book/source/sec_seq-induction.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 36,922 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_gt-intro">
<title>Problems and Definitions</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ol>
<li>
<p>
... | discrete-book/source/sec_gt-intro.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.001 | true | false | 67,634 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<worksheet xmlns:xi="http://www.w3.org/2001/XInclude" margin="1in" bottom="1.25cm" xml:id="PA-coloring-vertices">
<title>Preview Activity</title>
<introduction>
<p>
For each graph below:
<ul>
<li>
<p>
Find a proper vertex coloring using some number of co... | discrete-book/source/activities/coloring-vertices.ptx | math | discrete-book | discrete-book/source/activities | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 7,106 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xml:id="sec_structures-relations" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Relations</title>
<subsection xml:id="subsec-introduction">
<title>Introduction</title>
<!-- Add investigate -->
<p>
How do we represent the relationships between mathematical ... | discrete-book/source/sec_structures-relations.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.02 | true | false | 2,751 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<appendix xml:id="peer-instruction" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Peer Instruction Questions</title>
<introduction>
<p> This appendix contains a collection of multiple choice questions focusing on the concepts from each section in the textbook. These ... | discrete-book/source/peer-instruction.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 2,120 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_counting-combine-outcomes">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Your Blu-ray collection consists of 9 comedies and 7 horror movies.
Give an example of a ... | discrete-book/source/exercises/counting-combine-outcomes.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 1,646 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_intro-structures">
<title>Discrete Structures</title>
<subsection>
<title>Introduction</title>
<investigation>
<p>
A double-six domino set consists of tiles containing pairs of numbers, each from 0 to 6.
... | discrete-book/source/sec_intro-structures.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.002 | true | false | 27,311 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<backmatter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="backmatter">
<title>Backmatter</title>
<!-- <xi:include href="peer-instruction.ptx" /> -->
<solutions divisional="hint">
<title>Selected Hints</title>
</solutions>
<solutions divisional="solution">
... | discrete-book/source/backmatter.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 617 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<pretext xmlns:xi="http://www.w3.org/2001/XInclude" xml:lang="en-US">
<xi:include href="./bookinfo.ptx" />
<book xml:id="dmoi4">
<title>Discrete Mathematics</title>
<subtitle>An Open Introduction, 4th Edition</subtitle>
<xi:include href="./frontmatter.ptx" />
<xi:include... | discrete-book/source/dmoi.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 670 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_gt-coloring">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
What is the smallest number of colors you need to properly color the vertices of <m>K_{4,5}</m>?
... | discrete-book/source/exercises/gt-coloring.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 18,353 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<exercises xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="exercises_seq-polynomial">
<title>Additional Exercises</title>
<exercise>
<statement>
<p>
Your friendly neighborhood bodega has a candy machine that gives 7 Skittles to the first customer who pu... | discrete-book/source/exercises/seq-polynomial.ptx | math | discrete-book | discrete-book/source/exercises | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 13,915 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_gt-conc">
<title>Chapter Summary</title>
<p>
Hopefully this chapter has given you some sense of the wide variety of graph theory topics as well as why these studies are interesting.
There are many more in... | discrete-book/source/sec_gt-conc.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 2,859 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<worksheet xmlns:xi="http://www.w3.org/2001/XInclude" margin="1in" bottom="1.25cm" xml:id="Activity-13-combinatorial-proofs">
<title>Preview Activity</title>
<page>
<introduction>
<p>
It is often possible to find the answer to a counting question in two dif... | discrete-book/source/activities/combinatorial-proofs.ptx | math | discrete-book | discrete-book/source/activities | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 5,520 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xml:id="sec_logic-structures" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Proofs about Discrete Structures</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<... | discrete-book/source/sec_logic-structures.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0 | true | false | 38,978 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_comb-proofs">
<title>Combinatorial Proofs</title>
<idx> <h>combinatorial proof</h> </idx>
<idx> <h>proof</h> <h>combinatorial</h> </idx>
<objectives>
<introduction>
<p>
After completing thi... | discrete-book/source/sec_counting-proofs.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.007 | true | false | 34,732 | -1 |
\begin{tikzpicture}[scale=1.5]
\node[circle,draw] (a) at (0,2) {\tiny Al};
\node[circle,draw] (b) at (1,2) {\tiny Bob};
\node[circle,draw] (c) at (2,2) {\tiny Cat};
\node[circle,draw] (d) at (3,2) {\tiny Dirk};
\node[circle,draw] (e) at (4,2) {\tiny Eva};
\node[circle,draw] (C) at (.5,0) {\tiny Calc};
\no... | discrete-book/source/assets/class-relation.tex | math | discrete-book | discrete-book/source/assets | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0 | true | false | 592 | -1 |
%%% This is a set of styles to include in the preamble of the generated latex code for the book Discrete Mathematics: an Open Introduction.
\usepackage{bold-extra}
\usepackage{marvosym} %for stop signs.
\usepackage{textcomp}
\usepackage{multicol}
% % FONT OPTIONS (pick one group to uncomment):
%newpx is my curren... | discrete-book/assets/latex-preamble-styles-exercises.tex | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0.455 | true | false | 3,680 | -1 |
<?xml version="1.0" encoding="UTF-8"?>
<section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_logic-proofs">
<title>Proofs</title>
<objectives>
<introduction>
<p>
After completing this section, you should be able to do the following.
</p>
</introduction>
<ul>
<li>
<p>
Identify t... | discrete-book/source/sec_logic-proofs.ptx | math | discrete-book | discrete-book | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | other | false | 0.003 | true | false | 58,171 | -1 |
\begin{tikzpicture}[scale=1.5]
\foreach \i in {1,...,6} {
\node[circle, draw] (a\i) at (120-\i*60:1) {\tiny \i};
\draw[->] (a\i) edge [in=100-\i*60, out=140-\i*60,looseness=12] (a\i);
}
\draw[->] (a2) edge (a1);
\draw[->] (a3) edge (a1);
\draw[->] (a4) edge (a1);
\draw[->] (a4) edge (a2);
... | discrete-book/source/assets/dir-multigraph-relation.tex | math | discrete-book | discrete-book/source/assets | CC-BY-NC-SA-4.0 | CC-BY-NC-SA-4.0 | 2d28f5dee1370beeed729d3712d53638597885dd | doc | false | 0 | true | false | 457 | -1 |
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