+++
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: $f^{-1}(\mathbb A)$