• 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:

      • $(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$
      • $(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$

    • ${ Ai }{i \in \mathbb I}$ 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: $f^{-1}(\mathbb A)$