From 89f574766493c257d54a20d84ab9549009f25d10 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 25 Oct 2022 21:48:37 +0000 Subject: [PATCH 01/61] init fabio-lama notes --- .../student-notes/fabio-lama/README.md | 64 +++++++++++++++ .../student-notes/fabio-lama/build.sh | 77 +++++++++++++++++++ 2 files changed, 141 insertions(+) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/README.md create mode 100755 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/build.sh diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/README.md b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/README.md new file mode 100644 index 00000000..deeefc5f --- /dev/null +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/README.md @@ -0,0 +1,64 @@ +# 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. + +* TODO + +# Building + +_NOTE_: This step is only necessary if you chose to modify the base docments. + +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 offences. + +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, especially on the books: + +* TODO + +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.** diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/build.sh b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/build.sh new file mode 100755 index 00000000..309f406f --- /dev/null +++ b/level-4/fundamentals-of-computer-science/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 ${FILE} + + # Convert HTML to PNG. + PDF_OUT="cheatsheet_${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 \ No newline at end of file From 6ef39677aa63104e0ef022297720cd525ff41911 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 25 Oct 2022 22:14:27 +0000 Subject: [PATCH 02/61] start section on propositional logic and syntax --- .../fabio-lama/src/propositional_logic.adoc | 38 +++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc new file mode 100644 index 00000000..e7512c52 --- /dev/null +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc @@ -0,0 +1,38 @@ += Cheatsheet - Propositional Logic +Fabio Lama +:description: Module: CM1025 Fundamentals to Computer Science, started 25. October 2022 +:doctype: article +:sectnums: 4 +:stem: + +== Intro + +A proposition is a statement that can be either _true_ or _false_. It must be +one or the other, and it cannot be both. + +== Syntax + +Propositions are denoted by capital letters, such as stem:[P, Q, ...]. General +statements are denoted by lowercase letters, such as stem:[p, q, ...] + +=== Connectives + +**Logical NOT**: stem:[not p] is true if and only if stem:[p] is false (also +called _negation_). + +**Logical OR**: stem:[p vv q] is true if and only if at least one of stem:[p] or +stem:[q] is true or if both stem:[p] and stem:[q] are true (also called +_disjunction_). + +**Logical AND**: stem:[p ^^ q] is true if and only if both stem:[p] and stem:[q] +are true (also called _conjunction_). + +**Logical IF...THEN**: stem:[p -> q] is true if and only if either stem:[p] is +false or stem:[q] is true (also called _conditional_ or _implication_). stem:[p] +is the premise, stem:[q] is the conclusion. + +**Logical IF and only IF**: stem:[p harr q] is true if and only if both stem:[p] +and stem:[q] are true (also called _bi-conditional_). + +**Exclusive OR: XOR**: stem:[p o+ q] is true if stem:[p] or stem:[q] is true but +not both. From 500a0ad0251231827e2dd9b629d998ea1e067cdc Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 25 Oct 2022 22:29:15 +0000 Subject: [PATCH 03/61] add translation on connectives section --- .../fabio-lama/src/propositional_logic.adoc | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc index e7512c52..139400a1 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc @@ -36,3 +36,22 @@ and stem:[q] are true (also called _bi-conditional_). **Exclusive OR: XOR**: stem:[p o+ q] is true if stem:[p] or stem:[q] is true but not both. + +==== Translation to Connectives + +As an example, lets consider the propositions: + +* stem:[P = "I study 20 hours a week"] +* stem:[R = "I will pass the exam"] +* stem:[S = "I will be happy"] +* stem:[Q = "I attend all the lectures"] + +And the following connectives: + +[stem] +++++ +(P vv Q) -> (R ^^ S) +++++ + +which is a translation of: "**If** I study 20 hours a week **or** attend all +the lectures, **then** I will pass the exam **and** I will be happy." From 040bd78b00a352c29dbe9e991107056a5453f4e8 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Tue, 25 Oct 2022 23:36:39 +0000 Subject: [PATCH 04/61] add truth tables section and operator precedence --- .../fabio-lama/src/propositional_logic.adoc | 83 ++++++++++++++++++- 1 file changed, 81 insertions(+), 2 deletions(-) diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc index 139400a1..d70aaca5 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc @@ -15,7 +15,7 @@ one or the other, and it cannot be both. Propositions are denoted by capital letters, such as stem:[P, Q, ...]. General statements are denoted by lowercase letters, such as stem:[p, q, ...] -=== Connectives +== Connectives **Logical NOT**: stem:[not p] is true if and only if stem:[p] is false (also called _negation_). @@ -37,7 +37,7 @@ and stem:[q] are true (also called _bi-conditional_). **Exclusive OR: XOR**: stem:[p o+ q] is true if stem:[p] or stem:[q] is true but not both. -==== Translation to Connectives +=== Translation to Connectives As an example, lets consider the propositions: @@ -55,3 +55,82 @@ And the following connectives: which is a translation of: "**If** I study 20 hours a week **or** attend all the lectures, **then** I will pass the exam **and** I will be happy." + +== Truth Tables + +=== Negation: stem:[not] + +[stem] +++++ +true = not false\ +false = not true +++++ + +=== Conjunction: stem:[^^] + +[stem] +++++ +true ^^ true = true\ +true ^^ false = false\ +false ^^ true = false\ +false ^^ false = false +++++ + +=== Disjunction: stem:[vv] + +[stem] +++++ +true vv true = true\ +true vv false = true\ +false vv true = true\ +false vv false = false +++++ + +=== Implication: stem:[->] + +[stem] +++++ +true -> true = true\ +true -> false = false\ +false -> true = true \ +false -> false = true +++++ + +TODO: Clarify this. + +=== Bi-conditional: stem:[harr] + +[stem] +++++ +1 harr 1 = 1\ +1 harr 0 = 0\ +0 harr 1 = 0\ +0 harr 0 = 1 +++++ + +=== Exclusive or: XOR, stem:[o+] + +[stem] +++++ +1 o+ 1 = 0\ +1 o+ 0 = 1\ +0 o+ 1 = 1\ +0 o+ 0 = 0 +++++ + +== Operator Precedence + +Operators are applied in the following order (ascending): + +. stem:[not] +. stem:[^^] +. stem:[vv] +. stem:[->] +. stem:[harr] + +For example: + +[stem] +++++ +p -> p ^^ not q vv s -= (p -> ((p ^^ (not q)) vv s)) +++++ \ No newline at end of file From f8cc7e527888bd5422caf94d8e797728d23aea52 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Wed, 26 Oct 2022 00:01:10 +0000 Subject: [PATCH 05/61] reference stackexchange link for implication --- .../fabio-lama/src/propositional_logic.adoc | 14 +++++++++++++- 1 file changed, 13 insertions(+), 1 deletion(-) diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc index d70aaca5..66bd9778 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc @@ -96,7 +96,19 @@ false -> true = true \ false -> false = true ++++ -TODO: Clarify this. +==== +NOTE: This can seem weird at first, this answer helps: https://math.stackexchange.com/a/100288 + +> If you start out with a false premise, then, as far as implication is +concerned, you are free to conclude anything. (This corresponds to the fact +that, when stem:[p] is false, the implication stem:[p -> q] is true no matter +what stem:[q] is.) + +> If you start out with a true premise, then the implication should be true only +when the conclusion is also true. (This corresponds to the fact that, when +stem:[p] is true, the truth of the implication is the same as the truth of +stem:[q].) +==== === Bi-conditional: stem:[harr] From b8a21d892e529a609d62e0f6add0bad2ae976fe4 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Wed, 26 Oct 2022 00:02:04 +0000 Subject: [PATCH 06/61] add chapter on tautology and consistency --- .../fabio-lama/src/tautology_and_consistency.adoc | 6 ++++++ 1 file changed, 6 insertions(+) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc new file mode 100644 index 00000000..114dd338 --- /dev/null +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc @@ -0,0 +1,6 @@ += Cheatsheet - Tautology and Consistency +Fabio Lama +:description: Module: CM1025 Fundamentals to Computer Science, started 25. October 2022 +:doctype: article +:sectnums: 4 +:stem: From 00c2772e872eddd97303cdc998230367f4a6c82c Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sat, 29 Oct 2022 12:47:32 +0000 Subject: [PATCH 07/61] add first-order logic document --- .../fabio-lama/src/first-order_logic.adoc | 64 +++++++++++++++++++ .../fabio-lama/src/propositional_logic.adoc | 28 +++++++- .../src/tautology_and_consistency.adoc | 6 -- 3 files changed, 91 insertions(+), 7 deletions(-) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc delete mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc new file mode 100644 index 00000000..3cd336c1 --- /dev/null +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc @@ -0,0 +1,64 @@ += Cheatsheet - First-Order Logic +Fabio Lama +:description: Module: CM1025 Fundamentals to Computer Science, started 25. October 2022 +:doctype: article +:sectnums: 4 +:stem: + +== Predicates + +**Predicates** describe properties of objects. + +For example: + +[stem] +++++ +"odd"(3) +++++ + +stem:["odd"(3)] means stem:[3] is an odd number. stem:["odd"] is a predicate, +stem:[3] is an object. Predicates take arguments and become **propositions**. + +Connectives can be applied: + +[stem] +++++ +"odd"(3) ^^ "prime"(3) +++++ + +This means that stem:[3] is odd but also prime. + +== Quantifiers + +We use the symbol stem:[EE] to indicate the existence of something +(**existential quantifier**). + +[stem] +++++ +EE x " odd"(x) +++++ + +This means that there exists some stem:[x] that is odd. + +We denote the **univeral quantifier** as stem:[AA]. + +[stem] +++++ +AA x ("odd"(x) vv "even"(x)) +++++ + +This meant that **for all** stem:[x] the number is either even or odd. + +Other examples, "All Ps are Qs": + +[stem] +++++ +AA x (P(x) -> Q(x)) +++++ + +And "No Ps are Qs": + +[stem] +++++ +AA x (P(x) -> not Q(x)) +++++ diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc index 66bd9778..1caa67a6 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/propositional_logic.adoc @@ -145,4 +145,30 @@ For example: [stem] ++++ p -> p ^^ not q vv s -= (p -> ((p ^^ (not q)) vv s)) -++++ \ No newline at end of file +++++ + +== Descriptors + +A formula that's always true is called a **tautology**. A formula that is true +for at least one scenario is **consistent**. A formula that's never true is +**inconsistent**. A formula can also result in a **contradiction**. + +== Equivalances + +Formulas are equivalanent if they result in the same logical outcomes. + +For example (_De Morgan's Laws_): + +[stem] +++++ +not (p ^^ q) -= not p vv not q\ +not (p vv q) -= not p ^^ not q +++++ + +By using the first formula, we can see: + +[stem] +++++ +not (true ^^ true) = not true = false\ +not true vv not true = false vv false = false +++++ diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc deleted file mode 100644 index 114dd338..00000000 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/tautology_and_consistency.adoc +++ /dev/null @@ -1,6 +0,0 @@ -= Cheatsheet - Tautology and Consistency -Fabio Lama -:description: Module: CM1025 Fundamentals to Computer Science, started 25. October 2022 -:doctype: article -:sectnums: 4 -:stem: From 6d9ae3fbbc9f343109d9251b7397f8d8e90d4de1 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sat, 29 Oct 2022 13:00:25 +0000 Subject: [PATCH 08/61] add section on Quantifiers to Connectives --- .../fabio-lama/src/first-order_logic.adoc | 21 +++++++++++++++++++ 1 file changed, 21 insertions(+) diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc index 3cd336c1..e0385000 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/first-order_logic.adoc @@ -62,3 +62,24 @@ And "No Ps are Qs": ++++ AA x (P(x) -> not Q(x)) ++++ + +== Quantifiers to Connectives + +stem:[EE x, P(x)] where stem:[x in {x_1, x_2, ..., x_n}] means that there exists +some stem:[x] for which stem:[P(x)] is true. + +Denoted alternatively: + +[stem] +++++ +EE x, P(x) -= P(x_1) vv P(x_2) vv ... vv P(x_3) +++++ + +We can also conclude: + +[stem] +++++ +not EE x, P(x) -= not(P(x_1) vv P(x_2) vv ... vv P(x_3))\ +not EE x, P(x) -= not P(x_1) ^^ not P(x_2) ^^ ... ^^ not P(x_3)\ +not EE x, P(x) -= AA x, not P(x) +++++ From 16dd9ada13a433fbd0a7761edf30f07a2a0d2f46 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sat, 29 Oct 2022 14:09:18 +0000 Subject: [PATCH 09/61] add chapter on proofs --- .../student-notes/fabio-lama/src/proofs.adoc | 82 +++++++++++++++++++ 1 file changed, 82 insertions(+) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc new file mode 100644 index 00000000..75068dbc --- /dev/null +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc @@ -0,0 +1,82 @@ += Cheatsheet - Proofs +Fabio Lama +:description: Module: CM1025 Fundamentals to Computer Science, started 25. October 2022 +:doctype: article +:sectnums: 4 +:stem: + +== Intro + +A **proof** is a sequence of logical statements that explain why a statement is +true (or not true). However, some statements are true even if they cannot be +logically proven. + +== Direct Proof + +Direct proofs are easy because no particular technique is used. But it can be +difficult to find a starting point. + +Lets consider the example: + +[stem] +++++ +"even"(n) ^^ "even"(m) -> "even"(n+m) +++++ + +I.e. if both numbers are even, then the sum of both numbers is even, too. + +We can prove this: knowing that when an integer is even, it is twice another +integer. We conclude: + +[stem] +++++ +n = 2k, m = 2l\ +m+n = 2k+2l = 2(k+l) " (factored)"\ +k+l = t " (arbitrary name)"\ +m+n = 2t +++++ + +== Proof by Contradiction (Indirect Proof) + +Lets say we want to prove that a statement is true. We then assume that that +statement is false. Proof by contradiction means we now have to arrive at a +statement that contradicts our assumption (implying that the original statement is +true). + +NOTE: The following example is from Quora, https://qr.ae/pvqpcu + +For example, consider the statement _"there is no largest number in the world"_. +We now assume the opposite: _"stem:[N] is the largest number in the world"_. +What happens if we multiply that number by itself? + +[stem] +++++ +N xx N = N +++++ + +_..._ because we assume stem:[N] is the largest number in the world, meaning +stem:[NxxN] cannot be bigger than stem:[N]. + +We know that when we divide a number by itself, the result is stem:[1]: + +[stem] +++++ +N/N = 1 +++++ + +Respectively: + +[stem] +++++ +(N xx N)/N = (N xx cancel(N))/cancel(N) = N = 1 +++++ + +This implies that the largest number in the world is stem:[1], but we can prove +by contradiction that: + +[stem] +++++ +2 > 1 +++++ + +Hence the original statement _"there is not largest number in the world"_ is correct. \ No newline at end of file From 3a906912be2445f25efd4c44039b90c616df331a Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sat, 29 Oct 2022 14:36:04 +0000 Subject: [PATCH 10/61] expand proof by contrapositive --- .../student-notes/fabio-lama/src/proofs.adoc | 48 ++++++++++++++++++- 1 file changed, 46 insertions(+), 2 deletions(-) diff --git a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc index 75068dbc..97da1195 100644 --- a/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc +++ b/level-4/fundamentals-of-computer-science/student-notes/fabio-lama/src/proofs.adoc @@ -43,7 +43,7 @@ statement is false. Proof by contradiction means we now have to arrive at a statement that contradicts our assumption (implying that the original statement is true). -NOTE: The following example is from Quora, https://qr.ae/pvqpcu +NOTE: The following example is from Quora: https://qr.ae/pvqpcu For example, consider the statement _"there is no largest number in the world"_. We now assume the opposite: _"stem:[N] is the largest number in the world"_. @@ -79,4 +79,48 @@ by contradiction that: 2 > 1 ++++ -Hence the original statement _"there is not largest number in the world"_ is correct. \ No newline at end of file +Hence the original statement _"there is not largest number in the world"_ is correct. + +== Proof by Contrapositive (Indirect Proof) + +Lets say we want to prove the following statement: + +[stem] +++++ +A -> B " is true" +++++ + +The **contrapositive** being: + +[stem] +++++ +A -> B -= not B -> not A +++++ + +Now prove that stem:[not B -> not A] is true. Sometimes it's easier to prove the +contrapositive of a statement. + +NOTE: The following example is from Stackexchange: https://math.stackexchange.com/q/88565 + +For example: + +[stem] +++++ +x^2 != x -> x != 1 +++++ + +is equivalent to: + +[stem] +++++ +x = 1 -> x^2 = x +++++ + +Respectively, we can prove: + +[stem] +++++ +1^2 = 1 +++++ + +Hence, the original statement is true. From 6caaae1aa854bba7f8b5ae8e3b35ad9e38c13092 Mon Sep 17 00:00:00 2001 From: lamafab <42901763+lamafab@users.noreply.github.com> Date: Sat, 29 Oct 2022 15:22:53 +0000 Subject: [PATCH 11/61] fix typos, generate PDFs --- .../cheatsheet_first-order_logic.pdf | Bin 0 -> 630760 bytes .../fabio-lama/cheatsheet_proofs.pdf | Bin 0 -> 1310164 bytes .../cheatsheet_propositional_logic.pdf | Bin 0 -> 1560657 bytes .../fabio-lama/src/first-order_logic.adoc | 2 +- .../student-notes/fabio-lama/src/proofs.adoc | 2 +- 5 files changed, 2 insertions(+), 2 deletions(-) create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/cheatsheet_first-order_logic.pdf create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/cheatsheet_proofs.pdf create mode 100644 level-4/fundamentals-of-computer-science/student-notes/fabio-lama/cheatsheet_propositional_logic.pdf diff --git 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