diff --git a/config.toml b/config.toml index 9d12426..e9300f7 100644 --- a/config.toml +++ b/config.toml @@ -1,6 +1,6 @@ baseURL = "https://022385.xyz/" languageCode = "en-us" -title = "My New Hugo Site" +title = "abdulocracy" baseurl = "/" theme = "terminal" paginate = 5 @@ -9,39 +9,39 @@ paginate = 5 # dir name of your blog content (default is `content/posts`) contentTypeName = "posts" # ["orange", "blue", "red", "green", "pink"] - themeColor = "orange" + themeColor = "pink" # if you set this to 0, only submenu trigger will be visible - showMenuItems = 2 + showMenuItems = 0 # show selector to switch language showLanguageSelector = false # set theme to full screen width fullWidthTheme = false # center theme with default width - centerTheme = false + centerTheme = true # set a custom favicon (default is a `themeColor` square) - # favicon = "favicon.ico" + favicon = "favicon.png" [languages] [languages.en] languageName = "English" - title = "Terminal" - subtitle = "A simple, retro theme for Hugo" + title = "abdulocracy's personal site" + subtitle = "" keywords = "" - copyright = "" - menuMore = "Show more" - readMore = "Read more" - readOtherPosts = "Read other posts" + copyright = "© Abdulkadir Furkan Şanlı 2019" + menuMore = "≡" + readMore = "read more" + readOtherPosts = "read other posts" [languages.en.params.logo] - logoText = "Terminal" + logoText = "abdulocracy" logoHomeLink = "/" [languages.en.menu] [[languages.en.menu.main]] identifier = "about" - name = "About" + name = "about" url = "/about" [[languages.en.menu.main]] - identifier = "showcase" - name = "Showcase" - url = "/showcase" + identifier = "uni-notes" + name = "university notes" + url = "/tags/university-notes" diff --git a/content/about.md b/content/about.md new file mode 100644 index 0000000..10556a9 --- /dev/null +++ b/content/about.md @@ -0,0 +1,13 @@ +--- +title: "about" +date: 2019-11-04T11:14:55+01:00 +draft: false +--- + + + +- name: Abdulkadir Furkan Şanlı +- handle: abdulocracy +- contact: + - email: my handle at disroot dot org + - irc (freenode): abdulocracy diff --git a/content/face.jpg b/content/face.jpg new file mode 100644 index 0000000..b854367 Binary files /dev/null and b/content/face.jpg differ diff --git a/content/favicon.png b/content/favicon.png new file mode 100644 index 0000000..72c0900 Binary files /dev/null and b/content/favicon.png differ diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark index f71910c..945e504 100644 --- a/content/posts/eidma.mmark +++ b/content/posts/eidma.mmark @@ -1,15 +1,12 @@ +++ title = "Introduction to Discrete Mathematics" -date = "2019-03-26T08:47:11+01:00" -author = "abdul" +date = "2019-11-04" +author = "Abdulkadir" showFullContent = false -tags = ["university-notes", ""] +tags = ["university-notes"] markup = "mmark" +++ -Discrete mathematics -==================== - - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. ## Propositional calculus @@ -18,17 +15,21 @@ Discrete mathematics - We should avoid variables when forming statements, as they may change the logical value. - $$2=7$$ statement - $$x=5$$ not a statement + - 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.). - When doing logic, we use propositional variables (e.g. p, q, r). - Can be either **true** or **false**. + - 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 $$(0001)$$ - 2. 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 $$(1011)$$) + - Conjunction: $$p \land q$$ is only true if both p and q are true $$(0001)$$ + - Disjunction: $$p \lor q$$ is only false if both p and q are false $$(0111)$$ + - Implication (material conditional): $$p \implies q$$ is false only if p is true and q is false (truth table $$(1011)$$) - $$\equiv \neg p \lor q$$ + - 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. - e.g. $$\neg[(p \lor q)] \land r$$ - $$(p \land)$$ is not a formula, as $$\land$$ requires 2 variables. @@ -38,6 +39,7 @@ Discrete mathematics - Distributivity: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$ - Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions. - $$\neg(B \lor C)$$ can be written as $$\neg B \land \neg C$$ + - Double negation law: $$\neg(\neg p) \equiv p$$ - De Morgan's laws: $$\neg(p \land q) \equiv \neg p \lor \neg q$$ and $$\neg(p \lor q) \equiv \neg p \land \neg q$$. @@ -46,8 +48,10 @@ Discrete mathematics - $$(p \implies q) \equiv (\neg q \implies \neg p)$$ prove by contraposition - $$(p \implies q) \equiv (\neg p \lor q)$$ - $$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$$ + - Contradiction law: - $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$ + - Tautology: $$\phi (p, q, ... r)$$ is a tautology *iff* $$\phi \equiv 1$$ ## Sets @@ -55,23 +59,27 @@ Discrete mathematics - $$2^\mathbb X = \{ A : A \subseteq \mathbb X\}$$ - $$2^\mathbb X = P(\mathbb X)$$ - All 2 object subsets of $$\mathbb X$$: $$P_2(\mathbb X)$$ + - $$A \subset B \equiv$$ every element of A is an element of B $$\equiv \{x \in \mathbb X : x \in A \implies x \in B\}$$ - Operations on sets: - Union - $$\cup$$ - $$A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}$$ - Intersection - $$\cap$$ - $$A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}$$ - Complement - $$A'$$ - $$A' = \{ x \in \mathbb X : \neg (x \in A) \}$$ - If $$x = \{ 1 \}$$ then $$x' = \emptyset$$ + - Equality of sets: $$A = B$$ iff $$x \in \mathbb X : (x \in A \iff x \in B)$$ - Difference of sets: - $$A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'$$ - Symmetric difference: $$A \div B = (A \setminus B) \cup (B \setminus A)$$ + - Laws of set algebra: - 1. $$A \cup B = B \cup A , A \cap B = B \cap A$$ - 2. $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$ - 3. $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa - 4. $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$ - 5. $$(A \cup B)' = A' \cap B'$$ vice versa - 6. $$A \cup A' = \mathbb X, A \cap A' = \emptyset$$ + - $$A \cup B = B \cup A , A \cap B = B \cap A$$ + - $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$ + - $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa + - $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$ + - $$(A \cup B)' = A' \cap B'$$ vice versa + - $$A \cup A' = \mathbb X, A \cap A' = \emptyset$$ + - Note: $$\{ \emptyset \} \neq \emptyset$$, one is a set with one element, one is the empty set, no elements ($$\{ \}$$) - Quip: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$ @@ -80,22 +88,27 @@ Discrete mathematics - $$\forall$$ means "for all" and $$\exists$$ means "there exists" - $$\forall$$ - Shorthand for $$\land$$ e.g. $$(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$$ + - $$\exists$$ - 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 - 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 - $$(\forall x) (\exists y) x > y$$ is true - shortcut: $$(\exists x, y) \equiv (\exists x) (\exists y)$$ + - Quantifiers can be expressed in set language, sort of a definition in terms of sets: - $$(\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}$$ - $$(\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset$$ - $$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$$ + - Order of quantifiers matters. ## Relations - Cartesian product: - $$A \times B = \{ (p, q) : p \in A \land q \in B \}$$ + - Def: A relation $$R$$ on a set $$\mathbb X$$ is a subset of $$\mathbb X \times \mathbb X$$ ($$R \subseteq \mathbb X \times \mathbb X$$) - Graph of a function $$f()$$: $$\{ (x, f(x) : x \in Dom(f) \}$$ - Properties of: @@ -103,17 +116,21 @@ Discrete mathematics 2. Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$ 3. Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$ 4. Antisymmetricity: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$ + - Equivalence relations: - 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: $$p R q \equiv n | p - q$$ - 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 $$a$$. - $$(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$$ + - 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: - $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$$ - $$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$$ + - $$\{ A_i \}_{i \in \mathbb I}$$ is a partition *iff* there exists an equivalence relation $$R$$ on $$\mathbb X$$ such that: - $$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$$ - $$(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$$ + - The quotient set: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$ diff --git a/layouts/partials/extended_footer.html b/layouts/partials/extended_footer.html index 7eaa536..52cb203 100644 --- a/layouts/partials/extended_footer.html +++ b/layouts/partials/extended_footer.html @@ -2,6 +2,8 @@ To add an extended footer section, please create `layouts/partials/extended_footer.html` in your Hugo directory. --> + + + + + +{{ partial "extended_footer.html" . }} +