From 835828a38ea8548144fc3b7d6bc820d2f8fb513c Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?= Date: Thu, 21 Nov 2019 20:51:06 +0100 Subject: [PATCH] Switch back to KaTeX, with Pandoc markup MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Abdulkadir Furkan Şanlı --- content/posts/eidma.md | 1 + layouts/partials/extended_footer.html | 5 +- public/about/index.html | 19 +- public/categories/index.html | 19 +- public/index.html | 23 +- public/index.xml | 4 +- public/posts/eidma/index.html | 365 +++++++++--------------- public/posts/index.html | 23 +- public/posts/index.xml | 4 +- public/tags/index.html | 19 +- public/tags/university-notes/index.html | 23 +- public/tags/university-notes/index.xml | 4 +- 12 files changed, 198 insertions(+), 311 deletions(-) diff --git a/content/posts/eidma.md b/content/posts/eidma.md index a38feb7..408a2e9 100644 --- a/content/posts/eidma.md +++ b/content/posts/eidma.md @@ -2,6 +2,7 @@ title = "Introduction to Discrete Mathematics" date = "2019-11-20" tags = ["university-notes"] +markup = "pandoc" +++ - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. diff --git a/layouts/partials/extended_footer.html b/layouts/partials/extended_footer.html index 009877a..aa29bc4 100644 --- a/layouts/partials/extended_footer.html +++ b/layouts/partials/extended_footer.html @@ -3,7 +3,7 @@ To add an extended footer section, please create `layouts/partials/extended_footer.html` in your Hugo directory. --> - ---> + diff --git a/public/about/index.html b/public/about/index.html index 3b53ce4..2894e15 100644 --- a/public/about/index.html +++ b/public/about/index.html @@ -181,17 +181,16 @@ + + + + - - diff --git a/public/categories/index.html b/public/categories/index.html index 7fcaf96..6f673f6 100644 --- a/public/categories/index.html +++ b/public/categories/index.html @@ -164,17 +164,16 @@ + + + + - - diff --git a/public/index.html b/public/index.html index 000e0fa..b77cab9 100644 --- a/public/index.html +++ b/public/index.html @@ -159,9 +159,7 @@
- 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.
@@ -205,17 +203,16 @@ + + + + - - diff --git a/public/index.xml b/public/index.xml index 9d32b79..0a267de 100644 --- a/public/index.xml +++ b/public/index.xml @@ -18,9 +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 169904c..d8bfb9f 100644 --- a/public/posts/eidma/index.html +++ b/public/posts/eidma/index.html @@ -6,9 +6,7 @@ - + @@ -31,9 +29,7 @@ - + @@ -42,9 +38,7 @@ - + @@ -151,314 +145,224 @@
- - -
    +
    • 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.

      - +
    • 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 $\equiv$.

      • - -

    • 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).

      - +
    • 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.

      - +
    • 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:

      - +
    • 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.

      - +
    • 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): $p \iff p \equiv (p \implies q) \land (q \implies p)$

      • - -

    • Contraposition law:

      - +
    • 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:

      - +
    • 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 $\equiv {x \in \mathbb X : x \in A \implies x \in B}$

      • - -

    • Operations on sets:

      - +
    • \(A \subset B \equiv\) 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 $x \in \mathbb X : (x \in A \iff x \in B)$

      • - -

    • Difference of sets:

      - +
    • 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:

      - +
    • 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

      • - -

    • With quantifiers we can write logical statements e.g.

      - +
    • \(\neg \forall \equiv \exists\), 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:

      - +
    • 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.

      • +
    • Order of quantifiers matters.
    -

    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 $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)

      • - -

    • Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$

      • - -

    • Properties of:

      - +
    • 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:

      - +
    • 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

      - +
    • 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:
          -
        • $(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$
          • -
        • $(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$
          • +
        • \((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)
          • +
        • \((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)
        • - -

      • ${ Ai }{i \in \mathbb I}$ 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

      - +
    • 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: \(f^{-1}(\mathbb A)\)
@@ -490,17 +394,16 @@ + + + + - - diff --git a/public/posts/index.html b/public/posts/index.html index 31e89ea..648fc9f 100644 --- a/public/posts/index.html +++ b/public/posts/index.html @@ -156,9 +156,7 @@
- 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.
@@ -202,17 +200,16 @@ + + + + - - diff --git a/public/posts/index.xml b/public/posts/index.xml index ea5687e..737366e 100644 --- a/public/posts/index.xml +++ b/public/posts/index.xml @@ -18,9 +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 bcc4cd2..709f098 100644 --- a/public/tags/index.html +++ b/public/tags/index.html @@ -164,17 +164,16 @@ + + + + - - diff --git a/public/tags/university-notes/index.html b/public/tags/university-notes/index.html index 2f533dc..10153be 100644 --- a/public/tags/university-notes/index.html +++ b/public/tags/university-notes/index.html @@ -156,9 +156,7 @@
- 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.
@@ -202,17 +200,16 @@ + + + + - - diff --git a/public/tags/university-notes/index.xml b/public/tags/university-notes/index.xml index ee4b26b..f8316aa 100644 --- a/public/tags/university-notes/index.xml +++ b/public/tags/university-notes/index.xml @@ -18,9 +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.