From 11d8fc4e549228a7615b1da3eaad356814c4a445 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?= Date: Sat, 7 Dec 2019 17:34:42 +0100 Subject: [PATCH] Add link to music server to menu MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Abdulkadir Furkan Şanlı --- config.toml | 4 + public/about/index.html | 8 + public/categories/index.html | 8 + public/index.html | 10 +- public/index.xml | 2 +- public/posts/eidma/index.html | 232 ++++++++++++------------ public/posts/index.html | 10 +- public/posts/index.xml | 2 +- public/tags/index.html | 8 + public/tags/university-notes/index.html | 10 +- public/tags/university-notes/index.xml | 2 +- 11 files changed, 178 insertions(+), 118 deletions(-) diff --git a/config.toml b/config.toml index aee2007..7969c48 100644 --- a/config.toml +++ b/config.toml @@ -45,3 +45,7 @@ paginate = 5 identifier = "uni-notes" name = "university notes" url = "/tags/university-notes" + [[languages.en.menu.main]] + identifier = "music" + name = "music" + url = "https://music.abdulocra.cy" diff --git a/public/about/index.html b/public/about/index.html index 67bf811..e57bb2b 100644 --- a/public/about/index.html +++ b/public/about/index.html @@ -91,6 +91,10 @@ +
  • music
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  • university notes
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  • music
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  • university notes
    • diff --git a/public/categories/index.html b/public/categories/index.html index 6f673f6..d979e04 100644 --- a/public/categories/index.html +++ b/public/categories/index.html @@ -93,6 +93,10 @@ +
  • music
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  • university notes
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  • music
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  • university notes
    • diff --git a/public/index.html b/public/index.html index 907c1e8..10c1868 100644 --- a/public/index.html +++ b/public/index.html @@ -94,6 +94,10 @@ +
  • music
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  • university notes
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  • music
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  • university notes
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    - 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. + 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/index.xml b/public/index.xml index 3bc39a1..d7702d7 100644 --- a/public/index.xml +++ b/public/index.xml @@ -18,7 +18,7 @@ Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/posts/eidma/ - 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. + 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 index c337591..7425352 100644 --- a/public/posts/eidma/index.html +++ b/public/posts/eidma/index.html @@ -6,7 +6,7 @@ - + @@ -29,7 +29,7 @@ - + @@ -38,7 +38,7 @@ - + @@ -91,6 +91,10 @@ +
  • music
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  • university notes
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  • music
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  • university notes
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  • 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
      • +
    • 2=7 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.).
  • When doing logic, we use propositional variables (e.g. p, q, r).
      @@ -164,107 +172,107 @@
  • The operations done on propositional variables are called propositional connectives.
      -
    • 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 \((1011)\)) +
    • 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))
        -
      • \(\equiv \neg p \lor q\)
        • +
      • \equiv \neg p \lor q
  • Not necessarily connectives but unary operations:
      -
    • 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.
      • -
    • 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: \(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)
      • +
    • 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 \(\neg B \land \neg C\)
        • +
      • \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): \(p \iff p \equiv (p \implies q) \land (q \implies p)\)
    • +
  • 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 q \implies \neg p) prove by contraposition
        -
      • \((p \implies q) \equiv (\neg p \lor q)\)
        • -
      • \((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)
        • +
      • (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 \(p \land \neg p \equiv 0\)
      • +
    • 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

      -
    • We will consider subsets of universal set \(\mathbb X\) +
    • 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 \(\mathbb X\): \(P_2(\mathbb X)\)
        • +
      • 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 \(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)
      • +
    • 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 - \(A'\) - \(A' = \{ x \in \mathbb X : \neg (x \in A) \}\) +
      • 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 \(x' = \emptyset\)
          • +
        • 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:
        -
      • \(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\)
        • -
      • Symmetric difference: \(A \div B = (A \setminus B) \cup (B \setminus A)\)
        • +
      • 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:
        -
      • \(A \cup B = B \cup A , A \cap B = B \cap A\)
        • -
      • \((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)
        • -
      • \((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) 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
        • -
      • \(A \cup A' = \mathbb X, A \cap A' = \emptyset\)
        • +
      • 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: \(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)
      • +
    • 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”
      • -
    • \(\forall\): +
    • \phi - prepositional function: yields only true or false value
      • +
    • \forall means “for all” and \exists means “there exists”
      • +
    • \forall:
        -
      • Shorthand for \(\land\) e.g. \((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)
        • +
      • 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.
        -
      • \((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\) is a statement and is false
        • -
      • \((\forall x) (\exists y) x > y\) is true
        • -
      • shortcut: \((\exists x, y) \equiv (\exists x) (\exists y)\)
        • +
      • (\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:
        -
      • \((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)
        • -
      • \((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)
        • -
      • \((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)
        • +
      • (\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.
    @@ -272,117 +280,117 @@
    • 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 \(\mathbb X \times \mathbb X\) (\(R \subseteq \mathbb X \times \mathbb X\))
      • -
    • Graph of a function \(f()\): \(\{ (x, f(x) : x \in Dom(f) \}\)
      • +
    • 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:
        -
      • Reflexivity: \((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)
        • -
      • Symmetricity: \([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)
        • -
      • Transitivity: \((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)
        • -
      • Antisymmetricity: \((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)
        • +
      • Reflexivity: (\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x
        • +
      • 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) ]
        • +
      • Transitivity: (\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)
        • +
      • 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 \([x]_R = \{ y \in \mathbb X : x R y \}\) +
      • 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 \(a\).
          • -
        • \((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)
          • +
        • 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: +
      • 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:
          -
        • \((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)
          • -
        • \((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)
          • +
        • (\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: +
      • \{ A_i \}_{i \in \mathbb I} is a partition iff there exists an equivalence relation R on \mathbb X such that:
          -
        • \((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)
          • -
        • \((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)
          • +
        • (\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: \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)
        • +
      • The quotient set: \mathbb X / R = \{ [a] : a \in \mathbb X \}

    Posets

    • Partial orders
        -
      • \(\mathbb X\) is a set, \(R \subseteq \mathbb X \times \mathbb X\)
        • -
      • Def: \(R\) is a partial order on \(\mathbb X\) iff \(R\) is: +
      • \mathbb X is a set, R \subseteq \mathbb X \times \mathbb X
        • +
      • Def: R is a partial order on \mathbb X iff R is:
        • Reflexive
        • Antisymmetric
        • Transitive
        • -
      • Def: \(m \in \mathbb X\) is said to be: +
      • Def: m \in \mathbb X is said to be:
          -
        • maximal element in \((\mathbb X, \preccurlyeq)\) iff \((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)
          • -
        • largest iff \((\forall a \in \mathbb X) (a \preccurlyeq m)\)
          • -
        • minimal iff \((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)
          • -
        • smallest iff \((\forall a \in \mathbb X) (m \preccurlyeq a)\)
          • +
        • maximal element in (\mathbb X, \preccurlyeq) iff (\forall a \in \mathbb X) m \preccurlyeq a \implies m = a
          • +
        • largest iff (\forall a \in \mathbb X) (a \preccurlyeq m)
          • +
        • minimal iff (\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)
          • +
        • smallest iff (\forall a \in \mathbb X) (m \preccurlyeq a)
        • -
      • Def: A partial order \(R\) on \(\mathbb X\) is said to be “total” iff \((\forall x, y \in \mathbb X) (x R y \lor y R x)\)
        • -
      • Def: A subset \(B\) of \(\mathbb X\) is called a chain “chain” iff \(B\) is totally ordered by \(R\) +
      • Def: A partial order R on \mathbb X is said to be “total” iff (\forall x, y \in \mathbb X) (x R y \lor y R x)
        • +
      • Def: A subset B of \mathbb X is called a chain “chain” iff B is totally ordered by R
          -
        • \(C(\mathbb X)\) - the set of all chains in \((\mathbb X, R)\)
          • -
        • A chain \(D\) in \((\mathbb X, R)\) is called a maximal chain iff \(D\) is a maximal element in \((C(\mathbb X), R)\)
          • -
        • \(K \subseteq \mathbb X\) is called an antichain in \((\mathbb X, R)\) iff \((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)
          • -
        • Def: \(R\) is a partial order on \(\mathbb X\), \(R\) is called a well order iff \(R\) is a total order on \(X\) and every nonempty subset \(A\) of \(\mathbb X\) has the smallest element
          • +
        • C(\mathbb X) - the set of all chains in (\mathbb X, R)
          • +
        • A chain D in (\mathbb X, R) is called a maximal chain iff D is a maximal element in (C(\mathbb X), R)
          • +
        • K \subseteq \mathbb X is called an antichain in (\mathbb X, R) iff (\forall p, q \in K) (p R q \lor q R p \implies p = q)
          • +
        • Def: R is a partial order on \mathbb X, R is called a well order iff R is a total order on X and every nonempty subset A of \mathbb X has the smallest element

    Induction

      -
    • If \(\phi\) is a propositional function defined on \(\mathbb N\), if: +
    • If \phi is a propositional function defined on \mathbb N, if:
        -
      • \(\phi(1)\)
        • -
      • \((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)
        • -
      • \((\forall k \geq 1) \phi(k)\)
        • +
      • \phi(1)
        • +
      • (\forall n \geq 1) \phi(n) \implies \phi(n+1)
        • +
      • (\forall k \geq 1) \phi(k)

    Functions

      -
    • \(f: \mathbb X \to \mathbb Y\)
      • -
    • Def: \(f \subseteq \mathbb X \times \mathbb Y\) is said to be a function if: +
    • f: \mathbb X \to \mathbb Y
      • +
    • Def: f \subseteq \mathbb X \times \mathbb Y is said to be a function if:
        -
      • \((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)
        • -
      • \((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)
        • +
      • (\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))
        • +
      • (\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)
      • -
    • Types of functions \(f: \mathbb X \to \mathbb Y\): +
    • Types of functions f: \mathbb X \to \mathbb Y:
        -
      • \(f\) is said to be an injection ( 1 to 1 function) iff \((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)
        • -
      • \(f\) is said to be a surjection (onto function) iff \((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)
        • -
      • If \(f^{-1}\) is a function from \(\mathbb Y \to \mathbb X\) then \(f^{-1}\) is called the inverse function for \(f\) +
      • f is said to be an injection ( 1 to 1 function) iff (\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)
        • +
      • f is said to be a surjection (onto function) iff (\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y
        • +
      • If f^{-1} is a function from \mathbb Y \to \mathbb X then f^{-1} is called the inverse function for f
          -
        • Fact: \(f^{-1}\) is a function iff \(f\) is a bijection (1 to 1 and onto)
          • +
        • Fact: f^{-1} is a function iff f is a bijection (1 to 1 and onto)
      • -
    • For some set \(\mathbb A\) the image of \(\mathbb A\) by \(f\) is \(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\). We can also define the inverse of an image even when the function itself isn’t invertible: \(f^{-1}(\mathbb A)\)
      • +
    • For some set \mathbb A the image of \mathbb A by f is f(\mathbb A) = \{ f(x) : x \in \mathbb A \}. We can also define the inverse of an image even when the function itself isn’t invertible: f^{-1}(\mathbb A)

    Combinatorics

      -
    • \(|\mathbb A|\) size (number of elements) of \(\mathbb A\)
      • +
    • |\mathbb A| size (number of elements) of \mathbb A
    • Rule of addition:
        -
      • If \(\mathbb A, \mathbb B \subseteq \mathbb X\) and \(|\mathbb A|, |\mathbb B| \in \mathbb N\) and \(\mathbb A \cap \mathbb B = \emptyset\) then \(|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|\)
        • -
      • Can be generalized as: \[ +
      • If \mathbb A, \mathbb B \subseteq \mathbb X and |\mathbb A|, |\mathbb B| \in \mathbb N and \mathbb A \cap \mathbb B = \emptyset then |\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|
        • +
      • Can be generalized as: (\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\ |\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\ (\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset) -\]
        • +
    • Rule of multiplication:
        -
      • \(\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|\)
        • -
      • Can be generalized as: \[ +
      • \mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|
        • +
      • Can be generalized as: (\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\ |\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}| -\]
        • +
    diff --git a/public/posts/index.html b/public/posts/index.html index 648fc9f..061fce0 100644 --- a/public/posts/index.html +++ b/public/posts/index.html @@ -93,6 +93,10 @@ +
  • music
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    • @@ -111,6 +115,10 @@ +
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    - 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. + 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/index.xml b/public/posts/index.xml index 737366e..da049dc 100644 --- a/public/posts/index.xml +++ b/public/posts/index.xml @@ -18,7 +18,7 @@ Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/posts/eidma/ - 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. + 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/tags/index.html b/public/tags/index.html index 709f098..ddcb4c6 100644 --- a/public/tags/index.html +++ b/public/tags/index.html @@ -93,6 +93,10 @@ +
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    • diff --git a/public/tags/university-notes/index.html b/public/tags/university-notes/index.html index 10153be..2dd9c0d 100644 --- a/public/tags/university-notes/index.html +++ b/public/tags/university-notes/index.html @@ -93,6 +93,10 @@ +
  • music
    • + + +
  • university notes
    • @@ -111,6 +115,10 @@ +
  • music
    • + + +
  • university notes
    • @@ -156,7 +164,7 @@
    - 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. + 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/tags/university-notes/index.xml b/public/tags/university-notes/index.xml index f8316aa..917dc34 100644 --- a/public/tags/university-notes/index.xml +++ b/public/tags/university-notes/index.xml @@ -18,7 +18,7 @@ Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/posts/eidma/ - 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. + 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.