Do first proper configuration

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baseURL = "https://022385.xyz/" baseURL = "https://022385.xyz/"
languageCode = "en-us" languageCode = "en-us"
title = "My New Hugo Site" title = "abdulocracy"
baseurl = "/" baseurl = "/"
theme = "terminal" theme = "terminal"
paginate = 5 paginate = 5
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# dir name of your blog content (default is `content/posts`) # dir name of your blog content (default is `content/posts`)
contentTypeName = "posts" contentTypeName = "posts"
# ["orange", "blue", "red", "green", "pink"] # ["orange", "blue", "red", "green", "pink"]
themeColor = "orange" themeColor = "pink"
# if you set this to 0, only submenu trigger will be visible # if you set this to 0, only submenu trigger will be visible
showMenuItems = 2 showMenuItems = 0
# show selector to switch language # show selector to switch language
showLanguageSelector = false showLanguageSelector = false
# set theme to full screen width # set theme to full screen width
fullWidthTheme = false fullWidthTheme = false
# center theme with default width # center theme with default width
centerTheme = false centerTheme = true
# set a custom favicon (default is a `themeColor` square) # set a custom favicon (default is a `themeColor` square)
# favicon = "favicon.ico" favicon = "favicon.png"
[languages] [languages]
[languages.en] [languages.en]
languageName = "English" languageName = "English"
title = "Terminal" title = "abdulocracy's personal site"
subtitle = "A simple, retro theme for Hugo" subtitle = ""
keywords = "" keywords = ""
copyright = "" copyright = "© Abdulkadir Furkan Şanlı 2019"
menuMore = "Show more" menuMore = ""
readMore = "Read more" readMore = "read more"
readOtherPosts = "Read other posts" readOtherPosts = "read other posts"
[languages.en.params.logo] [languages.en.params.logo]
logoText = "Terminal" logoText = "abdulocracy"
logoHomeLink = "/" logoHomeLink = "/"
[languages.en.menu] [languages.en.menu]
[[languages.en.menu.main]] [[languages.en.menu.main]]
identifier = "about" identifier = "about"
name = "About" name = "about"
url = "/about" url = "/about"
[[languages.en.menu.main]] [[languages.en.menu.main]]
identifier = "showcase" identifier = "uni-notes"
name = "Showcase" name = "university notes"
url = "/showcase" url = "/tags/university-notes"

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---
title: "about"
date: 2019-11-04T11:14:55+01:00
draft: false
---
<image src="../face.jpg" width="173" height="150" />
- name: Abdulkadir Furkan Şanlı
- handle: abdulocracy
- contact:
- email: my handle at disroot dot org
- irc (freenode): abdulocracy

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+++ +++
title = "Introduction to Discrete Mathematics" title = "Introduction to Discrete Mathematics"
date = "2019-03-26T08:47:11+01:00" date = "2019-11-04"
author = "abdul" author = "Abdulkadir"
showFullContent = false showFullContent = false
tags = ["university-notes", ""] tags = ["university-notes"]
markup = "mmark" markup = "mmark"
+++ +++
Discrete mathematics
====================
- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
## Propositional calculus ## Propositional calculus
@ -18,17 +15,21 @@ Discrete mathematics
- We should avoid variables when forming statements, as they may change the logical value. - We should avoid variables when forming statements, as they may change the logical value.
- $$2=7$$ statement - $$2=7$$ statement
- $$x=5$$ not a statement - $$x=5$$ not a statement
- In logic we do not use the equals sign, we use the equivalence sign $$\equiv$$. - In logic we do not use the equals sign, we use the equivalence sign $$\equiv$$.
- Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.). - Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).
- When doing logic, we use propositional variables (e.g. p, q, r). - When doing logic, we use propositional variables (e.g. p, q, r).
- Can be either **true** or **false**. - Can be either **true** or **false**.
- The operations done on propositional variables are called propositional connectives. - The operations done on propositional variables are called propositional connectives.
1. Conjunction: $$p \land q$$ is only true if both p and q are true $$(0001)$$ - Conjunction: $$p \land q$$ is only true if both p and q are true $$(0001)$$
2. Disjunction: $$p \lor q$$ is only false if both p and q are false $$(0111)$$ - Disjunction: $$p \lor q$$ is only false if both p and q are false $$(0111)$$
3. Implication (material conditional): $$p \implies q$$ is false only if p is true and q is false (truth table $$(1011)$$) - Implication (material conditional): $$p \implies q$$ is false only if p is true and q is false (truth table $$(1011)$$)
- $$\equiv \neg p \lor q$$ - $$\equiv \neg p \lor q$$
- Not necessarily connectives but unary operations: - Not necessarily connectives but unary operations:
1. Negation: Denoted by ~, $$\neg$$ or NOT, negates the one input $$(10)$$. - Negation: Denoted by ~, $$\neg$$ or NOT, negates the one input $$(10)$$.
- A (propositional) formula is a "properly constructed" logical expression. - A (propositional) formula is a "properly constructed" logical expression.
- e.g. $$\neg[(p \lor q)] \land r$$ - e.g. $$\neg[(p \lor q)] \land r$$
- $$(p \land)$$ is not a formula, as $$\land$$ requires 2 variables. - $$(p \land)$$ is not a formula, as $$\land$$ requires 2 variables.
@ -38,6 +39,7 @@ Discrete mathematics
- Distributivity: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$ - Distributivity: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$
- Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions. - Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
- $$\neg(B \lor C)$$ can be written as $$\neg B \land \neg C$$ - $$\neg(B \lor C)$$ can be written as $$\neg B \land \neg C$$
- Double negation law: $$\neg(\neg p) \equiv p$$ - Double negation law: $$\neg(\neg p) \equiv p$$
- De Morgan's laws: $$\neg(p \land q) \equiv \neg p \lor \neg q$$ and $$\neg(p \lor q) \equiv \neg p \land \neg q$$. - De Morgan's laws: $$\neg(p \land q) \equiv \neg p \lor \neg q$$ and $$\neg(p \lor q) \equiv \neg p \land \neg q$$.
@ -46,8 +48,10 @@ Discrete mathematics
- $$(p \implies q) \equiv (\neg q \implies \neg p)$$ prove by contraposition - $$(p \implies q) \equiv (\neg q \implies \neg p)$$ prove by contraposition
- $$(p \implies q) \equiv (\neg p \lor q)$$ - $$(p \implies q) \equiv (\neg p \lor q)$$
- $$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$$ - $$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$$
- Contradiction law: - Contradiction law:
- $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$ - $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$
- Tautology: $$\phi (p, q, ... r)$$ is a tautology *iff* $$\phi \equiv 1$$ - Tautology: $$\phi (p, q, ... r)$$ is a tautology *iff* $$\phi \equiv 1$$
## Sets ## Sets
@ -55,23 +59,27 @@ Discrete mathematics
- $$2^\mathbb X = \{ A : A \subseteq \mathbb X\}$$ - $$2^\mathbb X = \{ A : A \subseteq \mathbb X\}$$
- $$2^\mathbb X = P(\mathbb X)$$ - $$2^\mathbb X = P(\mathbb X)$$
- All 2 object subsets of $$\mathbb X$$: $$P_2(\mathbb X)$$ - All 2 object subsets of $$\mathbb X$$: $$P_2(\mathbb X)$$
- $$A \subset B \equiv$$ every element of A is an element of B $$\equiv \{x \in \mathbb X : x \in A \implies x \in B\}$$ - $$A \subset B \equiv$$ every element of A is an element of B $$\equiv \{x \in \mathbb X : x \in A \implies x \in B\}$$
- Operations on sets: - Operations on sets:
- Union - $$\cup$$ - $$A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}$$ - Union - $$\cup$$ - $$A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}$$
- Intersection - $$\cap$$ - $$A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}$$ - Intersection - $$\cap$$ - $$A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}$$
- Complement - $$A'$$ - $$A' = \{ x \in \mathbb X : \neg (x \in A) \}$$ - Complement - $$A'$$ - $$A' = \{ x \in \mathbb X : \neg (x \in A) \}$$
- If $$x = \{ 1 \}$$ then $$x' = \emptyset$$ - If $$x = \{ 1 \}$$ then $$x' = \emptyset$$
- Equality of sets: $$A = B$$ iff $$x \in \mathbb X : (x \in A \iff x \in B)$$ - Equality of sets: $$A = B$$ iff $$x \in \mathbb X : (x \in A \iff x \in B)$$
- Difference of sets: - Difference of sets:
- $$A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'$$ - $$A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'$$
- Symmetric difference: $$A \div B = (A \setminus B) \cup (B \setminus A)$$ - Symmetric difference: $$A \div B = (A \setminus B) \cup (B \setminus A)$$
- Laws of set algebra: - Laws of set algebra:
1. $$A \cup B = B \cup A , A \cap B = B \cap A$$ - $$A \cup B = B \cup A , A \cap B = B \cap A$$
2. $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$ - $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$
3. $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa - $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa
4. $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$ - $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$
5. $$(A \cup B)' = A' \cap B'$$ vice versa - $$(A \cup B)' = A' \cap B'$$ vice versa
6. $$A \cup A' = \mathbb X, A \cap A' = \emptyset$$ - $$A \cup A' = \mathbb X, A \cap A' = \emptyset$$
- Note: $$\{ \emptyset \} \neq \emptyset$$, one is a set with one element, one is the empty set, no elements ($$\{ \}$$) - Note: $$\{ \emptyset \} \neq \emptyset$$, one is a set with one element, one is the empty set, no elements ($$\{ \}$$)
- Quip: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$ - Quip: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$
@ -80,22 +88,27 @@ Discrete mathematics
- $$\forall$$ means "for all" and $$\exists$$ means "there exists" - $$\forall$$ means "for all" and $$\exists$$ means "there exists"
- $$\forall$$ - $$\forall$$
- Shorthand for $$\land$$ e.g. $$(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$$ - Shorthand for $$\land$$ e.g. $$(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$$
- $$\exists$$ - $$\exists$$
- Shorthand for $$\lor$$ e.g. $$(\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5$$ - Shorthand for $$\lor$$ e.g. $$(\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5$$
- $$\neg \forall \equiv \exists$$, vice versa - $$\neg \forall \equiv \exists$$, vice versa
- With quantifiers we can write logical statements e.g. - With quantifiers we can write logical statements e.g.
- $$(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$$ is a statement and is false - $$(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$$ is a statement and is false
- $$(\forall x) (\exists y) x > y$$ is true - $$(\forall x) (\exists y) x > y$$ is true
- shortcut: $$(\exists x, y) \equiv (\exists x) (\exists y)$$ - shortcut: $$(\exists x, y) \equiv (\exists x) (\exists y)$$
- Quantifiers can be expressed in set language, sort of a definition in terms of sets: - Quantifiers can be expressed in set language, sort of a definition in terms of sets:
- $$(\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}$$ - $$(\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}$$
- $$(\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset$$ - $$(\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset$$
- $$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$$ - $$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$$
- Order of quantifiers matters. - Order of quantifiers matters.
## Relations ## Relations
- Cartesian product: - Cartesian product:
- $$A \times B = \{ (p, q) : p \in A \land q \in B \}$$ - $$A \times B = \{ (p, q) : p \in A \land q \in B \}$$
- Def: A relation $$R$$ on a set $$\mathbb X$$ is a subset of $$\mathbb X \times \mathbb X$$ ($$R \subseteq \mathbb X \times \mathbb X$$) - Def: A relation $$R$$ on a set $$\mathbb X$$ is a subset of $$\mathbb X \times \mathbb X$$ ($$R \subseteq \mathbb X \times \mathbb X$$)
- Graph of a function $$f()$$: $$\{ (x, f(x) : x \in Dom(f) \}$$ - Graph of a function $$f()$$: $$\{ (x, f(x) : x \in Dom(f) \}$$
- Properties of: - Properties of:
@ -103,17 +116,21 @@ Discrete mathematics
2. Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$ 2. Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$
3. Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$ 3. Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$
4. Antisymmetricity: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$ 4. Antisymmetricity: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$
- Equivalence relations: - Equivalence relations:
- Def: $$R \subseteq \mathbb X \times \mathbb X$$ is said to be an equivalence relation *iff* $$R$$ is reflexive, symmetric and transitive. - Def: $$R \subseteq \mathbb X \times \mathbb X$$ is said to be an equivalence relation *iff* $$R$$ is reflexive, symmetric and transitive.
- Congruence modulo n: $$p R q \equiv n | p - q$$ - Congruence modulo n: $$p R q \equiv n | p - q$$
- Def R - and equivalence relation of $$\mathbb X$$: The _equivalence class_ of an element $$x \in \mathbb X$$ is the set $$[x]_R = \{ y \in \mathbb X : x R y \}$$ - Def R - and equivalence relation of $$\mathbb X$$: The _equivalence class_ of an element $$x \in \mathbb X$$ is the set $$[x]_R = \{ y \in \mathbb X : x R y \}$$
- Every $$x \in \mathbb X$$ belongs to the equivalence class of some element $$a$$. - Every $$x \in \mathbb X$$ belongs to the equivalence class of some element $$a$$.
- $$(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$$ - $$(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$$
- Partitions - Partitions
- A partition is a set containing subsets of some set $$\mathbb X$$ such that their collective symmetric difference equals $$\mathbb X$$. A partition of is a set $$\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}$$ such that: - A partition is a set containing subsets of some set $$\mathbb X$$ such that their collective symmetric difference equals $$\mathbb X$$. A partition of is a set $$\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}$$ such that:
- $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$$ - $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$$
- $$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$$ - $$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$$
- $$\{ A_i \}_{i \in \mathbb I}$$ is a partition *iff* there exists an equivalence relation $$R$$ on $$\mathbb X$$ such that: - $$\{ A_i \}_{i \in \mathbb I}$$ is a partition *iff* there exists an equivalence relation $$R$$ on $$\mathbb X$$ such that:
- $$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$$ - $$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$$
- $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$$ - $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$$
- The quotient set: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$ - The quotient set: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$

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