Cheatsheets for CM1020 Discrete Mathemathics

level-4/discrete-mathematics/student-notes/fabio-lama/README.md

@@ -0,0 +1,78 @@
+# About
+
+Listed here is a collection of cheatsheet by topic. Those cheatsheets do not
+explain the topics in depth, but rather serve as quick lookup documents.
+Therefore, the course material provided by the lecturer should still be studied
+and understood. Not everything that is tested at the mid-terms or final exams is
+covered and the Author does not guarantee that the cheatsheets are free of
+errors.
+
+* [Set Theory](./cheatsheet_set_theory.pdf)
+* [Universal Set, Complement and Laws](./cheatsheet_universal_set_complement_laws.pdf)
+* [Functions](./cheatsheet_functions.pdf)
+* [Propositional & First-order Logic](/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/cheatsheet_propositional_logic.pdf)
+  * (covered in the Authors _Fundamentals of Computer Science_ module notes)
+* [Postulates of Boolean Algebra](./cheatsheet_postulates_boolean_algebra.pdf)
+* [Logic Gates](./cheatsheet_logic_gates.pdf)
+* [Proofs](/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/cheatsheet_proofs.pdf)
+  * (covered in the Authors _Fundamentals of Computer Science_ module notes)
+* [Graph Theory](./cheatsheet_graphs.pdf)
+* [Graph Theory: Isomorphism](./cheatsheet_graphs_isomorphism.pdf)
+* [Trees](./cheatsheet_trees.pdf)
+* [Relations](./cheatsheet_relations.pdf)
+* [Equivalence Relations & Classes](./cheatsheet_equivalence.pdf)
+* [Combinatorics](/level-4/computational-mathematics/student-notes/fabio-lama/cheatsheet_probability_combinatorics.pdf)
+  * (covered in the Authors _Computational Mathematics_ module notes)
+* [Binomial Coefficients & Identities](./cheatsheet_binomial_coefficients.pdf)
+
+# Building
+
+_NOTE_: This step is only necessary if you chose to modify the base documents.
+
+The base documents are written in [AsciiDoc](https://asciidoc.org/) and can be
+found in the `src/` directory.
+
+The following dependencies must be installed (Ubuntu):
+
+```console
+$ apt install -y ruby-dev wkhtmltopdf
+$ gem install asciidoctor
+$ chmod +x build.sh
+```
+
+To build the documents (PDF version):
+
+```console
+$ ./build.sh pdf
+```
+
+Optionally, for the HTML version:
+
+```console
+$ ./build.sh html
+```
+
+and for the PNG version:
+
+```console
+$ ./build.sh png
+```
+
+The generated output can be deleted with `./build.sh clean`.
+
+# Disclaimer
+
+The Presented Documents ("cheatsheets") by the Author ("Fabio Lama") are
+summaries of specific topics. The term "cheatsheet" implies that the Presented
+Documents are intended to be used as learning aids or as references for
+practicing and does not imply that the Presented Documents should be used for
+inappropriate practices during exams such as cheating or other offenses.
+
+The Presented Documents are heavily based on the learning material provided by
+the University of London, respectively the VLeBooks Collection database in the
+Online Library and the material provided on the Coursera platform.
+
+The Presented Documents may incorporate direct or indirect definitions,
+examples, descriptions, graphs, sentences and/or other content used in those
+provided materials. **At no point does the Author present the work or ideas
+incorporated in the Presented Documents as their own.**

level-4/discrete-mathematics/student-notes/fabio-lama/build.sh

@@ -0,0 +1,77 @@
+#!/bin/bash
+
+# Because `make` sucks.
+
+gen_html() {
+    # Remove suffix and prefix
+    FILE=$1
+    OUT=${FILE%.adoc}
+    HTML_OUT="cheatsheet_${OUT}.html"
+
+    asciidoctor $FILE -o ${HTML_OUT}
+}
+
+# Change directory to src/ in order to have images included correctly.
+cd "$(dirname "$0")/src/"
+
+case $1 in
+    html)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+        done
+
+        # Move up from src/
+        mv *.html ../
+        ;;
+    pdf)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PDF.
+            PDF_OUT="cheatsheet_${OUT}.pdf"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $PDF_OUT
+        done
+
+        # Move up from src/
+        mv *.pdf ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    png | img)
+        for FILE in *.adoc
+        do
+            # Generate HTML file.
+            gen_html ${FILE}
+
+            # Convert HTML to PNG.
+            IMG_OUT="cheatsheet_${OUT}.png"
+            wkhtmltopdf \
+                --enable-local-file-access \
+                --javascript-delay 2000\
+                $HTML_OUT $IMG_OUT
+        done
+
+        # Move up from src/
+        mv *.png ../
+
+        # Cleanup temporarily generated HTML files.
+        rm *.html > /dev/null 2>&1
+        ;;
+    clean)
+        rm *.html > /dev/null 2>&1
+        rm *.png > /dev/null 2>&1
+        rm ../*.html > /dev/null 2>&1
+        rm ../*.png > /dev/null 2>&1
+        ;;
+    *)
+        echo "Unrecognized command"
+        ;;
+esac

level-4/discrete-mathematics/student-notes/fabio-lama/src/binomial_coefficients.adoc

@@ -0,0 +1,74 @@
+= Cheatsheet - Binomial Coefficients & Identities
+Fabio Lama <fabio.lama@pm.me>
+:description: Module: CM1020- Discrete Mathematics, started 25. October 2022
+:doctype: article
+:sectnums: 4
+:stem:
+
+== Binomial Theorem
+
+An expression consisting of two terms, connected by a stem:[+] or stem:[-] sign,
+is called a **binomial expression**. As we increase the power of binomials,
+expanding them becomes more and more complicated:
+
+[stem]
+++++
+(x+y)^1 = x + y\
+(x+y)^2 = x^2 + 2xy + y^2\
+(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\
+...
+++++
+
+The **binomial theorem** helps us to simplify this expansion. Let stem:[x] and
+stem:[y] be variables, and stem:[n] a non-negative integer. The expansion of
+stem:[(x+y)^n] can be formalized as:
+
+[stem]
+++++
+(x+y)^n = sum_(k=0)^n ((n),(k)) x^k y^(n-k)
+++++
+
+The **binomial coefficients** are the coefficients in the binomial theorem and
+denoted as:
+
+[stem]
+++++
+((n),(k)) = (n!)/(k!(n-k)!)
+++++
+
+Here we say **"n choose k"**.
+
+For example:
+
+> What is the coefficient of stem:[x^8 y^7] in the expansion of stem:[(3x
+-y)^15].
+
+We can view the expression as stem:[(3x+ (-y))^15]. By the binomial theorem:
+
+[stem]
+++++
+(3x+ (-y))^15 = sum_(k=0)^15 ((15),(k)) (3x)^k (-y)^(15-k)
+++++
+
+The coefficient of stem:[x^8 y^7] in the expansion is obtained when stem:[k=8]:
+
+[stem]
+++++
+((15),(8)) (3)^8 (-1)^7 = -3^8 (15!)/(8!7!)
+++++
+
+=== Pascal's Identity
+
+If stem:[n] and stem:[k] are integers with stem:[n >= k >= 1], then:
+
+[stem]
+++++
+((n),(k)) + ((n),(k-1)) = ((n+1),(k))
+++++
+
+=== Pascal's Triangle
+
+**Pascals' triangle** is a number triangle with numbers arranged in staggered rows
+such that **stem:[a_(n,r)] is the binomial coefficient stem:[((n),(r))]**.
+
+image::./assets/pascals_triangle.png[align=center, width=600]

level-4/discrete-mathematics/student-notes/fabio-lama/src/equivalence.adoc

@@ -0,0 +1,61 @@
+= Cheatsheet - Equivalence Relations & Classes
+Fabio Lama <fabio.lama@pm.me>
+:description: Module: CM1020- Discrete Mathematics, started 25. October 2022
+:doctype: article
+:sectnums: 4
+:stem:
+
+NOTE: You should read the cheatsheet on _Relations_, too.
+
+== Definitions
+
+=== Equivalence Relation
+
+Let stem:[R] be a relation of elements on set stem:[S]. stem:[R] is an
+**equivalence relation** if and only if stem:[R] is **reflexive**, **symmetric**
+and **transitive**.
+
+=== Equivalence Classes
+
+Let stem:[R] be an **equivalence relation** on a set stem:[S]. Then, the
+**equivalence class** of stem:[a in S] is the **subset** of stem:[S] containing
+all the **elements related** to stem:[a] through stem:[R]:
+
+[stem]
+++++
+[a] = {x: x in S " and " xRa}
+++++
+
+For example:
+
+[stem]
+++++
+S = {1, 2, 3, 4, 5}\
+R = {(a, b) in S^2 | a - b " is an even number" }
+++++
+
+The set stem:[R] has two equivalence classes:
+
+[stem]
+++++
+[1] = [3] = [5] = {1, 3, 5}\
+[2] = [4] = {2, 4}
+++++
+
+== Partial & Total Order
+
+Let stem:[R] be a relation on elements in a set stem:[S]. stem:[R] is a
+**partial order** if and only if stem:[R] is **reflexive**, **anti-symmetric**
+and **transitive**.
+
+Additionally, stem:[R] is a **total** order if and only if:
+
+* stem:[R] is a **partial order**.
+* stem:[AA (a, b) in S] we have **either** stem:[aRb] **or** stem:[bRa].
+
+For example, the following relation is a total order:
+
+[stem]
+++++
+R = {(a, b) in ZZ^2 | a <= b}
+++++