# Preliminary notions

## Correspondence between equivalence relations and partitions

• Equivalence class: Given a set A, an equivalence relation R on A, and an element x of A, the equivalence class of x w.r.t. R is
[x]R = {y in A | x R y}
Quotient set: The quotient set of A w.r.t. R is
[A]R = {[x]R | x is in A }
• Partition: Given a set A, a partition P of A is a set of subsets of A such that
1. for every X, Y in P, X and Y are disjoint (i.e. their intersection is empty)
2. the union of al the sets in P is equal to A.

• Theorem (Correspondence between equivalence relations and partitions) Given an equivalence relation R on A, [A]R is a partition of A. Viceversa, if P is a partition of A, then the relation
R = {(x,y) | exists X in P s.t. x, y are in X }
is an equivalence relation on A (and the quotient of A w.r.t. R is exactly P).

## Classical Logic.

### Propositional calculus

#### Syntax

Formulas are expressions built on
• Propositional letters, p, q, r, ...
• Connectives: not, and, or, implies (->), logical equivalence (<->), ...

#### Meaning

The letters range on the set of boolean values B = {True, False} The connectives are functions on B. For instance: "p implies q" is the function (of p and q) which returns False when p is True and q is False, and True in all other cases.

Truth tables: a way to represent the meaning of each connective. The meaning (truth value) of more complex formulas can be derived by composing the truth tables.

Tautology: A formula which is True for all the possible truth values of the letters contained in it.

Examples:

• p -> p is a tautology
• p -> (p -> p) is a tautology
• (p -> p) -> p is not a tautology
• (p and q) -> p is a tautology
There are variuos systems to derive tautologies (i.e. to prove that a certain formula is a tautology), but usually the simplest method is to use the truth tables.