diff --git a/content/about.md b/content/about.md index e0d5e2e..5746660 100644 --- a/content/about.md +++ b/content/about.md @@ -1,7 +1,6 @@ --- title: "about" -date: 2019-11-04T11:14:55+01:00 -draft: false +date: 2019-11-04 --- diff --git a/content/posts/eidma.md b/content/posts/eidma.md new file mode 100644 index 0000000..a38feb7 --- /dev/null +++ b/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: $f^{-1}(\mathbb A)$ diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark deleted file mode 100644 index bcef92b..0000000 --- a/content/posts/eidma.mmark +++ /dev/null @@ -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: $$f^{-1}(\mathbb A)$$ diff --git a/layouts/partials/extended_footer.html b/layouts/partials/extended_footer.html index 0fbe96e..009877a 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. --> - + + + +