+++
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: $$\mathbb X / R = \{ [a] : a \in \mathbb X \}$$