Switch from KaTeX to MathJax, ditching mmark

Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>

content/about.md

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 ---
 title: "about"
-date: 2019-11-04T11:14:55+01:00
+date: 2019-11-04
-draft: false
 ---
 
 <image src="face.jpg" width="173" height="150" />

content/posts/eidma.md

@@ -0,0 +1,180 @@
++++
+title = "Introduction to Discrete Mathematics"
+date = "2019-11-20"
+tags = ["university-notes"]
++++
+
+- 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.
+	- Conjunction: $p \land q$ is only true if both p and q are true $(0001)$
+	- Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$
+	- Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
+		- $\equiv \neg p \lor q$
+
+- Not necessarily connectives but unary operations:
+	- 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: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$
+	- 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:
+	- $A \cup B = B \cup A , A \cap B = B \cap A$
+	- $(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$
+	- $(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ 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: $\{ x \in \mathbb R : x^2 = -1\} = \emptyset$
+
+## 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:
+	- 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 $[x]_R = \{ y \in \mathbb X : x R y \}$
+		- 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: $\mathbb X / R = \{ [a] : a \in \mathbb X \}$
+
+## 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:
+		- Reflexive
+		- Antisymmetric
+		- Transitive
+
+	- 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 $(\forall a \in \mathbb X) (m \preccurlyeq a)$
+
+	- 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
+
+## Induction
+
+- If $\phi$ is a propositional function defined on $\mathbb N$, if:
+   	- $\phi(1)$
+   	- $(\forall n \geq 1) \phi(n) \implies \phi(n+1)$
+	- $(\forall k \geq 1) \phi(k)$
+
+## Functions
+
+- $f: \mathbb X \to \mathbb Y$
+- Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:
+  - $(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$
+  - $(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$
+
+- 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 $f$
+      - 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: $f^{-1}(\mathbb A)$

content/posts/eidma.mmark

@@ -1,180 +0,0 @@
-+++
-title = "Introduction to Discrete Mathematics"
-date = "2019-11-20"
-tags = ["university-notes"]
-+++
-
-- 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.
-	- Conjunction: $$p \land q$$ is only true if both p and q are true $$(0001)$$
-	- Disjunction: $$p \lor q$$ is only false if both p and q are false $$(0111)$$
-	- Implication (material conditional): $$p \implies q$$ is false only if p is true and q is false (truth table $$(1011)$$)
-		- $$\equiv \neg p \lor q$$
-
-- Not necessarily connectives but unary operations:
-	- 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: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$
-	- 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:
-	- $$A \cup B = B \cup A , A \cap B = B \cap A$$
-	- $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$
-	- $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ 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: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$
-
-## 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:
-	- 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 $$[x]_R = \{ y \in \mathbb X : x R y \}$$
-		- 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: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$
-
-## 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:
-		- Reflexive
-		- Antisymmetric
-		- Transitive
-
-	- 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 $$(\forall a \in \mathbb X) (m \preccurlyeq a)$$
-
-	- 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
-
-## Induction
-
-- If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if:
-   	- $$\phi(1)$$
-   	- $$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$$
-	- $$(\forall k \geq 1) \phi(k)$$
-
-## Functions
-
-- $$f: \mathbb X \to \mathbb Y$$
-- Def: $$f \subseteq \mathbb X \times \mathbb Y$$ is said to be a function if:
-  - $$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$$
-  - $$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$$
-
-- 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 $$f$$
-      - 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: $$f^{-1}(\mathbb A)$$

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@@ -12,3 +12,15 @@ To add an extended footer section, please create
 <script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/contrib/auto-render.min.js"
     integrity="sha384-kWPLUVMOks5AQFrykwIup5lo0m3iMkkHrD0uJ4H5cjeGihAutqP0yW0J6dpFiVkI" crossorigin="anonymous"
     onload="renderMathInElement(document.body);"></script>
+-->
+
+<script>
+    MathJax = {
+        tex: {
+            inlineMath: [['$', '$'], ['\\(', '\\)']],
+            displayMath: [['$$', '$$'], ['\[', '\]']]
+        }
+    };
+</script>
+<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
+</script>