Introduction to Discrete Mathematics :: abdulocracy's personal site

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}|