From 8aa992c824529a31278140093a6ce11b9d04bdc2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?= Date: Mon, 4 Nov 2019 10:28:34 +0100 Subject: [PATCH] Regenerate site --- public/categories/index.html | 4 + public/index.html | 18 +- public/index.xml | 8 +- public/posts/eidma/index.html | 384 ++++++++++++++++++ public/posts/index.html | 18 +- public/posts/index.xml | 8 +- public/sitemap.xml | 16 +- public/tags/index.html | 4 + public/tags/index.xml | 10 + public/tags/university-notes/index.html | 203 +++++++++ public/tags/university-notes/index.xml | 24 ++ .../tags/university-notes/page/1/index.html | 1 + 12 files changed, 681 insertions(+), 17 deletions(-) create mode 100644 public/posts/eidma/index.html create mode 100644 public/tags/university-notes/index.html create mode 100644 public/tags/university-notes/index.xml create mode 100644 public/tags/university-notes/page/1/index.html diff --git a/public/categories/index.html b/public/categories/index.html index c574ff9..1eaa9c0 100644 --- a/public/categories/index.html +++ b/public/categories/index.html @@ -151,6 +151,10 @@ + + + diff --git a/public/index.html b/public/index.html index 8a63d14..3964894 100644 --- a/public/index.html +++ b/public/index.html @@ -128,7 +128,7 @@

- My First Post

+ Introduction to Discrete Mathematics
2019-03-26 @@ -138,19 +138,29 @@
+ + + #university-notes  + + #  + + +
+ Discrete mathematics Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. 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. +
Read more → + href="/posts/eidma/">Read more →
@@ -183,6 +193,10 @@ + + + diff --git a/public/index.xml b/public/index.xml index fb43992..becc354 100644 --- a/public/index.xml +++ b/public/index.xml @@ -12,12 +12,12 @@ - My First Post - https://022385.xyz/posts/my-first-post/ + Introduction to Discrete Mathematics + https://022385.xyz/posts/eidma/ Tue, 26 Mar 2019 08:47:11 +0100 - https://022385.xyz/posts/my-first-post/ - cunt + https://022385.xyz/posts/eidma/ + Discrete mathematics Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. 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. diff --git a/public/posts/eidma/index.html b/public/posts/eidma/index.html new file mode 100644 index 0000000..b961d76 --- /dev/null +++ b/public/posts/eidma/index.html @@ -0,0 +1,384 @@ + + + + + Introduction to Discrete Mathematics :: Terminal — A simple, retro theme for Hugo + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ Introduction to Discrete Mathematics

+
+ + + 2019-03-26 + + + + :: + abdul + + +
+ + + + + #university-notes  + + #  + + + + + + +
+

Discrete mathematics

+ +
    +
  • Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
    • +
+ +

Propositional calculus

+ +
    +
  • Comes from the linguistic concept that things can be either true or false.
    • +
  • 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. +
    2. Disjunction: \(p \lor q\) is only false if both p and q are false \((0111)\)
      3. +
    3. Implication (material conditional): \(p \implies q\) is false only if p is true and q is false (truth table \((1011)\)) + +
        +
      • \(\equiv \neg p \lor q\)
        • +
      4. +
    • +
  • Not necessarily connectives but unary operations: + +
      +
    1. Negation: Denoted by ~, \(\neg\) or NOT, negates the one input \((10)\).
      2. +
    • +
  • 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.
      • +
    • Logical equivalence: \(\phi(p, q, k) \equiv \psi(p, q, k)\), logical value of \(\phi\) is equal to logical value of \(\psi\).
      • +
    • Commutativity: \(p \land q \equiv q \land p\)
      • +
    • Associativity: \((p \land q) \land r \equiv p \land (q \land r)\)
      • +
    • 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. + +
        +
      • \(\neg(B \lor C)\) 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 \(\neg(p \lor q) \equiv \neg p \land \neg q\).

    • + +

  • If and only if (iff): \(p \iff p \equiv (p \implies q) \land (q \implies p)\)

    • + +

  • Contraposition law:

    + +
      +
    • \((p \implies q) \equiv (\neg q \implies \neg p)\) 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

+ +
    +
  • We will consider subsets of universal set \(\mathbb X\) + +
      +
    • \(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)
      • +
    • \(2^\mathbb X = P(\mathbb X)\)
      • +
    • 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. +
    2. \((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)
      3. +
    3. \((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) vice versa
      4. +
    4. \(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)
      5. +
    5. \((A \cup B)' = A' \cap B'\) vice versa
      6. +
    6. \(A \cup A' = \mathbb X, A \cap A' = \emptyset\)
      7. +
    • +
  • Note: \(\{ \emptyset \} \neq \emptyset\), one is a set with one element, one is the empty set, no elements (\(\{ \}\))
    • +
  • Quip: \(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)
    • +
+ +

Quantifiers

+ +
    +
  • \(\phi\) - prepositional function: yields only true or false value
    • +
  • \(\forall\) 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: + +
      +
    1. Reflexivity: \((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)
      2. +
    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. +
    3. Transitivity: \((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)
      4. +
    4. Antisymmetricity: \((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)
      5. +
    • +
  • 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: \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)
      • +
    • +
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- My First Post

+ Introduction to Discrete Mathematics
2019-03-26 @@ -135,19 +135,29 @@
+ + + #university-notes  + + #  + + +
+ Discrete mathematics Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. 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. +
Read more → + href="/posts/eidma/">Read more →
@@ -180,6 +190,10 @@ + + + diff --git a/public/posts/index.xml b/public/posts/index.xml index 9bdf606..c1a512f 100644 --- a/public/posts/index.xml +++ b/public/posts/index.xml @@ -12,12 +12,12 @@ - My First Post - https://022385.xyz/posts/my-first-post/ + Introduction to Discrete Mathematics + https://022385.xyz/posts/eidma/ Tue, 26 Mar 2019 08:47:11 +0100 - https://022385.xyz/posts/my-first-post/ - cunt + https://022385.xyz/posts/eidma/ + Discrete mathematics Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. 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. diff --git a/public/sitemap.xml b/public/sitemap.xml index 0d74299..0d589e6 100644 --- a/public/sitemap.xml +++ b/public/sitemap.xml @@ -3,7 +3,7 @@ xmlns:xhtml="http://www.w3.org/1999/xhtml"> - https://022385.xyz/posts/my-first-post/ + https://022385.xyz/posts/eidma/ 2019-03-26T08:47:11+01:00 @@ -12,17 +12,23 @@ 2019-03-26T08:47:11+01:00 + + https://022385.xyz/tags/ + 2019-03-26T08:47:11+01:00 + + https://022385.xyz/ 2019-03-26T08:47:11+01:00 + + https://022385.xyz/tags/university-notes/ + 2019-03-26T08:47:11+01:00 + + https://022385.xyz/categories/ - - https://022385.xyz/tags/ - - \ No newline at end of file diff --git a/public/tags/index.html b/public/tags/index.html index d7d1a16..085f669 100644 --- a/public/tags/index.html +++ b/public/tags/index.html @@ -151,6 +151,10 @@ + + + diff --git a/public/tags/index.xml b/public/tags/index.xml index fb9d35e..c13a46f 100644 --- a/public/tags/index.xml +++ b/public/tags/index.xml @@ -6,9 +6,19 @@ Recent content in Tags on Terminal Hugo -- gohugo.io en-us + Tue, 26 Mar 2019 08:47:11 +0100 + + university-notes + https://022385.xyz/tags/university-notes/ + Tue, 26 Mar 2019 08:47:11 +0100 + + https://022385.xyz/tags/university-notes/ + + + \ No newline at end of file diff --git a/public/tags/university-notes/index.html b/public/tags/university-notes/index.html new file mode 100644 index 0000000..c588ea1 --- /dev/null +++ b/public/tags/university-notes/index.html @@ -0,0 +1,203 @@ + + + + + university-notes :: Terminal — A simple, retro theme for Hugo + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ Introduction to Discrete Mathematics

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+ + 2019-03-26 + + :: + abdul +
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+ + + Discrete mathematics Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. 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. + + +
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