personal-site/content/posts/eidma.md

+++ 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)