Discrete Structures

# Lecture Notes, 15 Sep 2010

 Last Update: 15 September 2010 Note: or material is highlighted

1. Propositional Equivalences:

1. There are a whole host of famous propositional equivalences;
see the charts on pp. 24–25.

2. Among the most important are:

• (AB) ≡ (¬AB) ≡ (¬B → ¬A) ≡ ¬(A ∧ ¬B)
• the distributive laws (look them up!)
• De Morgan's Laws (look them up!)

Construct truth tables to convince yourself of these!

3. Also note that, because (AB) ≡ (¬AB),
it's also the case that ¬(AB) ≡ ¬(¬AB)
(You should convince yourself of this by constructing another truth table!)

2. Syntactic vs. Semantic Proofs:

1. There are 2 different ways of proving things:

1. semantically (in terms of truth values)
2. syntactically (by means of symbol manipulation)

• This is the way that computers would have to do it—by manipulating symbols,
which is what they do best

2. Here's a theorem for which I'll give 2 proofs, one semantic, one syntactic:

3. Theorem:
¬(AB) ≡ (A ∧ ¬B)

proof #1 (semantic, using the def of logical equiv):

Show: For all rows of their t-tables, tval(¬(A → B)) = tval(A ∧ ¬B)

A B (AB) ¬(A → B) ¬B (A ∧ ¬B)
T T T F F F
T F F T T T
F T T F F F
F F T F T F

Because the 4th and 6th columns are identical,
we can conclude that the 2 propositions are logically equivalent.
QED

Note that this symbol manipulation is a little bit like the Transformer toys that turn trucks into monsters, and vice versa.

Here, we "transformed" ¬(AB) into (A ∧ ¬B)
by manipulating the symbols according to the rules of Shakespeare's Law and propositional equivalences.

A semantic "proof" would be needed to decide the truth value of "John gave a book to Mary".

We need to look at the world to see if that sentence is true.

But we only need a syntactic "proof" to decide that "John gave a book to Mary" means the same thing as "Mary received a book from John".

We can determine that they are "equivalent in meaning" just by looking at the syntax of the sentences;
we don't have to look at the world

Clearly, you could write a computer program to do this.

3. §1.3: Predicates & Quantifiers

1. The language of propositional logic is very "coarse grained"

1. It allows us to talk about The True and The False
but not about objects and their properties & relations

2. E.g., can't show the relationship between:

• My Mac is a networked computer.
• Some Macs are networked computers.
• All Macs are networked computers.

In propositional logic, these would be represented by p, q, r

2. So we need an "object-oriented", finer-grained logical language

• It's called first-order predicate logic (or sometimes "first-order predicate calculus",
which has nothing to do with MTH 141)

• We'll call it "first-order logic", or FOL, for short.
Next lecture…