Lecture Notes, propositional equivalence.

1. Logical Equivalences

Some of these "logically equivalent" expressions have special names


Identity           P ^ T       <==>  P         % Rules w/ one proposition
Identity           P v F       <==>  P
Domination         P ^ F       <==>  F
Domination         P v T       <==>  T
Negation           P ^ ~P      <==>  F
Negation           P v ~P      <==>  T
Idempotent         P ^ P       <==>  P 
Idempotent         P v P       <==>  P 
Commutative Law    P ^ Q       <==>  Q ^ P     % Rules w/ two propositions
Commutative Law    P v Q       <==>  Q v P
DeMorgans Law    ~(P v Q)      <==>  (~P ^ ~Q)
DeMorgans Law    ~(P ^ Q)      <==>  (~P v ~Q)
Implies Rule       P --> Q     <==>  (~P v Q)
Contrapositive:    P --> Q     <==>  ~Q --> ~P
Absorption         P ^ (P v Q) <==>  P
Absorption         P v (P ^ Q) <==>  P
Associative Law    P v (Q v R) <==>  (P v Q) v R  % Rules w/ 3 propositions
Associative Law    P ^ (Q ^ R) <==>  (P ^ Q) ^ R
Distributive Law   P ^ (Q v R) <==>  (P ^ Q) v (P ^ R) 
Distributive Law   P v (Q ^ R) <==>  (P v Q) ^ (P v R)

2. Proofs

Using the "real" logical equivalences, we can PROVE other equivalences, example: P ^ (~P v Q) <==> P ^ Q For this first homework, you must do every proof one step at a time, in excruciating detail.

3. Tautology

When a propositional expression is equivalent to TRUE, we call it a TAUTOLOGY:

in a tautology, every possible possible way of assigning truth values to the propositions leads to an expression which is true.

Q --> (~P v Q) <==> T

4. Contradictions

When a propositional expression is equivalent to FALSE, we call it a CONTRADICTION.

in a contradiction, every possible possible way of assigning truth values to the propositions leads to an expression which is false.

5. Equivalence

Expressions A, B are equivalent iff A <--> B is a tautology.

6. Completeness of And, OR, NOT

Can define an arbitrary propositional function, by specifying its truth values for a whole column of a truth table. We now constructively prove that we can express any function using just and, or, and not.

7. Optimization of logical circuitry.

There are two things that one might want to minimize:

Criteria one: length of the formula. If we define new connectives, we could make the length of every possible formula with two propositional variables be just three characters long. How many connectives would we need to add?
Criteria two: number of different connectives. If we use fewer different kinds of connectives, we can perhaps simplify the manufacturing process and make it easier to build chips. Can we make up our logical formulas without using some of our connectives?

What do we have to do to prove that we don't need a specific connective?

Why do we use the set of connectives that we do?

Today we start with a few more examples of propositional equivalence. First, for practice, we prove one rule of equivalance as true.

Contrapositive: Prove this is logically equivalent...

  a --> b   <==>  ~b --> ~a       
  ~a v  b   <==>  ~b --> ~a       "implies rule"
  ~a v  b   <==>  ~~b v ~a        "implies rule"
  ~a v  b   <==>    b v ~a        "double negation"
  ~a v  b   <==>    ~a v b        "commutative rule"

Second, Can we add to the list of nice rules about "--->"? Prove or disprove the "associative rule of implication".

(p --> q) --> r  <==> p --> (q --> r)

~(~p v q) v r    ( we're just seeing if it works... so don't have to
                   do all the steps )

(p ^ ~q) v r     <==> ~p v (~q v r)

this looks alllll different. So find some way of assigning T,F to all the variables so that the two sides have different truth values.