Introduction to 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.

  • 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: \[ (\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\ |\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}| \]