Fall 2000, CSE 468:
Lecture 3 (Aug 30)
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
- for every X, Y in P, X and Y are disjoint (i.e. their intersection is empty)
- 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.