Switch from KaTeX to MathJax, ditching mmark
Signed-off-by: Abdulkadir Furkan Şanlı <abdulkadirfsanli@protonmail.com>
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Abdulkadir Furkan Şanlı
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---
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title: "about"
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date: 2019-11-04T11:14:55+01:00
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draft: false
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date: 2019-11-04
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---
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<image src="face.jpg" width="173" height="150" />
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180
content/posts/eidma.md
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180
content/posts/eidma.md
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title = "Introduction to Discrete Mathematics"
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date = "2019-11-20"
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tags = ["university-notes"]
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+++
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- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
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## Propositional calculus
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- Comes from the linguistic concept that things can be either true or false.
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- We should avoid variables when forming statements, as they may change the logical value.
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- $2=7$ statement
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- $x=5$ not a statement
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- In logic we do not use the equals sign, we use the equivalence sign $\equiv$.
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- 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).
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- Can be either **true** or **false**.
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||||
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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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- $\equiv \neg p \lor q$
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- Not necessarily connectives but unary operations:
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- Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.
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- A (propositional) formula is a "properly constructed" logical expression.
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- 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: $(p \land q) \land r \equiv p \land (q \land r)$
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- Distributivity: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$
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- Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
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- $\neg(B \lor C)$ can be written as $\neg B \land \neg C$
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- Double negation law: $\neg(\neg p) \equiv p$
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- De Morgan's laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.
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- If and only if (*iff*): $p \iff p \equiv (p \implies q) \land (q \implies p)$
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- Contraposition law:
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- $(p \implies q) \equiv (\neg q \implies \neg p)$ prove by contraposition
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- $(p \implies q) \equiv (\neg p \lor q)$
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- $(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$
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- Contradiction law:
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- $p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$
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- Tautology: $\phi (p, q, ... r)$ is a tautology *iff* $\phi \equiv 1$
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## Sets
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- We will consider subsets of universal set $\mathbb X$
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- $2^\mathbb X = \{ A : A \subseteq \mathbb X\}$
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- $2^\mathbb X = P(\mathbb X)$
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- All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$
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- $A \subset B \equiv$ every element of A is an element of B $\equiv \{x \in \mathbb X : x \in A \implies x \in B\}$
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- Operations on sets:
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- Union - $\cup$ - $A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}$
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- Intersection - $\cap$ - $A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}$
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- Complement - $A'$ - $A' = \{ x \in \mathbb X : \neg (x \in A) \}$
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- If $x = \{ 1 \}$ then $x' = \emptyset$
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- Equality of sets: $A = B$ iff $x \in \mathbb X : (x \in A \iff x \in B)$
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- Difference of sets:
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- $A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'$
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- Symmetric difference: $A \div B = (A \setminus B) \cup (B \setminus A)$
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- Laws of set algebra:
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- $A \cup B = B \cup A , A \cap B = B \cap A$
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- $(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$
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- $(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ vice versa
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- $A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$
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- $(A \cup B)' = A' \cap B'$ vice versa
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- $A \cup A' = \mathbb X, A \cap A' = \emptyset$
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- Note: $\{ \emptyset \} \neq \emptyset$, one is a set with one element, one is the empty set, no elements ($\{ \}$)
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- Quip: $\{ x \in \mathbb R : x^2 = -1\} = \emptyset$
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## Quantifiers
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- $\phi$ - prepositional function: yields only true or false value
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- $\forall$ means "for all" and $\exists$ means "there exists"
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- $\forall$:
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- Shorthand for $\land$ e.g. $(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$
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- $\exists$:
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- Shorthand for $\lor$ e.g. $(\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5$
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- $\neg \forall \equiv \exists$, vice versa
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- With quantifiers we can write logical statements e.g.
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- $(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$ is a statement and is false
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- $(\forall x) (\exists y) x > y$ is true
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- shortcut: $(\exists x, y) \equiv (\exists x) (\exists y)$
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- Quantifiers can be expressed in set language, sort of a definition in terms of sets:
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- $(\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}$
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- $(\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset$
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- $(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$
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- Order of quantifiers matters.
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## Relations
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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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- 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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- Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$
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- 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: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$
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- Equivalence relations:
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- Def: $R \subseteq \mathbb X \times \mathbb X$ is said to be an equivalence relation *iff* $R$ is reflexive, symmetric and transitive.
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- Congruence modulo n: $p R q \equiv n | p - q$
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- Def R - and equivalence relation of $\mathbb X$: The _equivalence class_ of an element $x \in \mathbb X$ is the set $[x]_R = \{ y \in \mathbb X : x R y \}$
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- Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.
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- $(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$
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- Partitions
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- 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:
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- $(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$
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- $(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$
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- $\{ A_i \}_{i \in \mathbb I}$ is a partition *iff* there exists an equivalence relation $R$ on $\mathbb X$ such that:
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- $(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$
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- $(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$
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- The quotient set: $\mathbb X / R = \{ [a] : a \in \mathbb X \}$
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## Posets
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- Partial orders
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- $\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$
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- Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:
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- Reflexive
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- Antisymmetric
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- Transitive
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- Def: $m \in \mathbb X$ is said to be:
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- maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$
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- largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$
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- minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$
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- smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$
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- Def: A partial order $R$ on $\mathbb X$ is said to be *"total"* iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$
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- Def: A subset $B$ of $\mathbb X$ is called a chain *"chain"* iff $B$ is totally ordered by $R$
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- $C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$
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- A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$
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- $K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$
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- 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
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## Induction
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- If $\phi$ is a propositional function defined on $\mathbb N$, if:
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- $\phi(1)$
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- $(\forall n \geq 1) \phi(n) \implies \phi(n+1)$
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- $(\forall k \geq 1) \phi(k)$
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## Functions
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- $f: \mathbb X \to \mathbb Y$
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- Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:
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- $(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$
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- $(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$
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- Types of functions $f: \mathbb X \to \mathbb Y$:
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- $f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$
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- $f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$
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- If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
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- Fact: $f^{-1}$ is a function iff $f$ is a *bijection* (1 to 1 and onto)
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- 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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@ -1,180 +0,0 @@
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+++
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title = "Introduction to Discrete Mathematics"
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date = "2019-11-20"
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tags = ["university-notes"]
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+++
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- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
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|
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## Propositional calculus
|
||||
|
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- Comes from the linguistic concept that things can be either true or false.
|
||||
- We should avoid variables when forming statements, as they may change the logical value.
|
||||
- $$2=7$$ statement
|
||||
- $$x=5$$ not a statement
|
||||
|
||||
- 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 $$\psi$$.
|
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- Commutativity: $$p \land q \equiv q \land p$$
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- Associativity: $$(p \land q) \land r \equiv p \land (q \land r)$$
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- Distributivity: $$p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$$
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- Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
|
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- $$\neg(B \lor C)$$ can be written as $$\neg B \land \neg C$$
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||||
|
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- Double negation law: $$\neg(\neg p) \equiv p$$
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- De Morgan's laws: $$\neg(p \land q) \equiv \neg p \lor \neg q$$ and $$\neg(p \lor q) \equiv \neg p \land \neg q$$.
|
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- If and only if (*iff*): $$p \iff p \equiv (p \implies q) \land (q \implies p)$$
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- Contraposition law:
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- $$(p \implies q) \equiv (\neg q \implies \neg p)$$ prove by contraposition
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- $$(p \implies q) \equiv (\neg p \lor q)$$
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- $$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$$
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- Contradiction law:
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- $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$
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- Tautology: $$\phi (p, q, ... r)$$ is a tautology *iff* $$\phi \equiv 1$$
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## Sets
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- We will consider subsets of universal set $$\mathbb X$$
|
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- $$2^\mathbb X = \{ A : A \subseteq \mathbb X\}$$
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- $$2^\mathbb X = P(\mathbb X)$$
|
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- All 2 object subsets of $$\mathbb X$$: $$P_2(\mathbb X)$$
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- $$A \subset B \equiv$$ every element of A is an element of B $$\equiv \{x \in \mathbb X : x \in A \implies x \in B\}$$
|
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- Operations on sets:
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- Union - $$\cup$$ - $$A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}$$
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- Intersection - $$\cap$$ - $$A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}$$
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- Complement - $$A'$$ - $$A' = \{ x \in \mathbb X : \neg (x \in A) \}$$
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- If $$x = \{ 1 \}$$ then $$x' = \emptyset$$
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|
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- 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:
|
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- $$A \cup B = B \cup A , A \cap B = B \cap A$$
|
||||
- $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$
|
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- $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa
|
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- $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$
|
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- $$(A \cup B)' = A' \cap B'$$ 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: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$
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||||
|
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## Quantifiers
|
||||
|
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- $$\phi$$ - prepositional function: yields only true or false value
|
||||
- $$\forall$$ means "for all" and $$\exists$$ means "there exists"
|
||||
- $$\forall$$:
|
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- Shorthand for $$\land$$ e.g. $$(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$$
|
||||
|
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- $$\exists$$:
|
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- 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:
|
||||
- $$(\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}$$
|
||||
- $$(\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset$$
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- $$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )$$
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- Order of quantifiers matters.
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## Relations
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|
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- 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: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$
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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: $$(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$$
|
||||
|
||||
- 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: $$f^{-1}(\mathbb A)$$
|
||||
@ -3,7 +3,7 @@ To add an extended footer section, please create
|
||||
`layouts/partials/extended_footer.html` in your Hugo directory.
|
||||
-->
|
||||
|
||||
<!-- KaTeX -->
|
||||
<!-- KaTeX
|
||||
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.css"
|
||||
integrity="sha384-zB1R0rpPzHqg7Kpt0Aljp8JPLqbXI3bhnPWROx27a9N0Ll6ZP/+DiW/UqRcLbRjq" crossorigin="anonymous">
|
||||
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.11.1/dist/katex.min.js"
|
||||
@ -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>
|
||||
|
||||
Loading…
Reference in New Issue
Block a user