200 lines
10 KiB
Markdown
200 lines
10 KiB
Markdown
---
|
|
title: Introduction to Discrete Mathematics
|
|
date: "2019-11-04"
|
|
markup: pandoc
|
|
---
|
|
|
|
- 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: $f^{-1}(\mathbb A)$
|
|
|
|
## Combinatorics
|
|
|
|
- $|\mathbb A|$ size (number of elements) of $\mathbb A$
|
|
- Rule of addition:
|
|
- If $\mathbb A, \mathbb B \subseteq \mathbb X$ and $|\mathbb A|, |\mathbb B| \in \mathbb N$ and $\mathbb A \cap \mathbb B = \emptyset$ then $|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|$
|
|
- Can be generalized as:
|
|
$$
|
|
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|
|
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
|
|
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
|
|
$$
|
|
- Rule of multiplication:
|
|
- $\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|$
|
|
- Can be generalized as:
|
|
$$
|
|
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\
|
|
|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}|
|
|
$$
|