diff --git a/content/posts/eidma.md b/content/posts/eidma.md
index a38feb7..408a2e9 100644
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+++ b/content/posts/eidma.md
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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
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`layouts/partials/extended_footer.html` in your Hugo directory.
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diff --git a/public/about/index.html b/public/about/index.html
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diff --git a/public/categories/index.html b/public/categories/index.html
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diff --git a/public/index.html b/public/index.html
index 000e0fa..b77cab9 100644
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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.
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diff --git a/public/index.xml b/public/index.xml
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--- a/public/index.xml
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@@ -18,9 +18,7 @@
Wed, 20 Nov 2019 00:00:00 +0000https://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
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Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
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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.).
-
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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.
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The operations done on propositional variables are called propositional connectives.
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The operations done on propositional variables are called propositional connectives.
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Conjunction: $p \land q$ is only true if both p and q are true $(0001)$
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Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$
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Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
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Conjunction: \(p \land q\) is only true if both p and q are true \((0001)\)
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Disjunction: \(p \lor q\) is only false if both p and q are false \((0111)\)
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Implication (material conditional): \(p \implies q\) is false only if p is true and q is false (truth table \((1011)\))
-
$\equiv \neg p \lor q$
+
\(\equiv \neg p \lor q\)
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-
Not necessarily connectives but unary operations:
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Not necessarily connectives but unary operations:
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Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.
+
Negation: Denoted by ~, \(\neg\) or NOT, negates the one input \((10)\).
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A (propositional) formula is a “properly constructed” logical expression.
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A (propositional) formula is a “properly constructed” logical expression.
-
e.g. $\neg[(p \lor q)] \land r$
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$(p \land)$ is not a formula, as $\land$ requires 2 variables.
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Logical equivalence: $\phi(p, q, k) \equiv \psi(p, q, k)$, logical value of $\phi$ is equal to logical value of $\psi$.
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Commutativity: $p \land q \equiv q \land p$
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Associativity: $(p \land q) \land r \equiv p \land (q \land r)$
\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)
-
-
Order of quantifiers matters.
+
Order of quantifiers matters.
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Relations
-
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Cartesian product:
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+
Cartesian product:
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$A \times B = { (p, q) : p \in A \land q \in B }$
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\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)
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Def: A relation $R$ on a set $\mathbb X$ is a subset of $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)
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Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$
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Properties of:
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+
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:
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Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$
-
Symmetricity: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$
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Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$
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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 $a$.
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:
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: \(f^{-1}(\mathbb A)\)
- 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.
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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 +0000https://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
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diff --git a/public/tags/university-notes/index.html b/public/tags/university-notes/index.html
index 2f533dc..10153be 100644
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+++ 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.
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diff --git a/public/tags/university-notes/index.xml b/public/tags/university-notes/index.xml
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--- 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 +0000https://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.