diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark new file mode 100644 index 0000000..f71910c --- /dev/null +++ b/content/posts/eidma.mmark @@ -0,0 +1,119 @@ ++++ +title = "Introduction to Discrete Mathematics" +date = "2019-03-26T08:47:11+01:00" +author = "abdul" +showFullContent = false +tags = ["university-notes", ""] +markup = "mmark" ++++ + +Discrete mathematics +==================== + +- 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. + 1. Conjunction: $$p \land q$$ is only true if both p and q are true $$(0001)$$ + 2. Disjunction: $$p \lor q$$ is only false if both p and q are false $$(0111)$$ + 3. 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: + 1. 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: + 1. $$A \cup B = B \cup A , A \cap B = B \cap A$$ + 2. $$(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$$ + 3. $$(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ vice versa + 4. $$A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$$ + 5. $$(A \cup B)' = A' \cap B'$$ vice versa + 6. $$A \cup A' = \mathbb X, A \cap A' = \emptyset$$ +- 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: + 1. Reflexivity: $$(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$$ + 2. Symmetricity: $$[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$$ + 3. Transitivity: $$(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$$ + 4. 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: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$ diff --git a/content/posts/my-first-post.md b/content/posts/my-first-post.md deleted file mode 100644 index c7c355f..0000000 --- a/content/posts/my-first-post.md +++ /dev/null @@ -1,8 +0,0 @@ -+++ -title = "My First Post" -date = "2019-03-26T08:47:11+01:00" -author = "abdul" -showFullContent = false -+++ - -cunt diff --git a/layouts/partials/extended_footer.html b/layouts/partials/extended_footer.html new file mode 100644 index 0000000..7eaa536 --- /dev/null +++ b/layouts/partials/extended_footer.html @@ -0,0 +1,8 @@ + + + +