Exercises for CS3511Week 31 (first week of practical)

Propositional Logic

Exercise 1

1. Express each formula using only (at most) the connectives listed. In each case use a truth table to prove the equivalence. (Note: is exclusive `or`)

a. Formula: pq. Connectives: {,}.b. Formula: pq. Connectives: {,,}. c. Formula: pq. Connectives: {, }.d. Formula: (pq) ((p)q). Conn: {,}.e. Formula: p. Conn: { | } (the Sheffer stroke).

Answer to Exercise 1. (Other answers possible)

a. Formula pq. Connectives: {,}.Answer: pqb. Formula: pq. Connectives: {,,}.Answer: (p q)(q p)c. Formula: pq. Connectives: {, }.Answer: (pq)(q p)d. Formula: (pq) ((p)q). Conn: {,}.Answer: q (This was a trick question, since you don’t need any

connectives.)

e. Formula: p. Conn: { | } (the Sheffer stroke).Answer: p|p

Ex. 2. Which of these are tautologies?

1. p (q p)2. p (p p)3. (q p) (p q) 4. (q p) (p q)5. (p (q r)) (q (p r))

Please prove your claims, using truth tables. (Hint: Ask what assignment of truth values to p,q, and r would falsify each formula. In this way you can disregard parts of the truth table).

Answer to Ex.2

1. p (q p) Tautologous2. p (p p) Tautologous3. (q p) (p q) Contingent4. (q p) (p q) Tautologous5. (p (q r)) (q (p r)) Tautologous

1,2,4,5 are known as “ ’paradoxes’ of implication”, because they contrast with implication in ordinary language.

Ex. 3a. Reading formulas off truth tables

• Background: In class, a proof was sketched for the claim that every propositional logic formula can be expressed using the connectives {, }. The proof proceeded essentially by “reading off” the correct formula off the truth table of any given formula.

• Task: Use this meticulous method to construct a formula equivalent to pq.

Answer to Ex. 3a. Steps:

1. Construct the truth table of pq.

2. Mark those two rows in the table that make pq TRUE.

3. Corresponding with these two rows, construct a disjunction of two formulas, one of which is (pq), and the other (qp).

4. Use the De Morgan Laws to convert this disjunction (pq)(qp) into the quivalent formula ((pq) (qp))

[5. Use truth tables again to check that these two formulas are indeed equivalent.]

Ex. 3b. Reading formulas off truth tables

• As Ex. 3, but with a difference:• Task: Use this meticulous method to

construct a formula equivalent to p|q.

• Question: Does this meticulous method always produce the shortest answer (i.e. the shortest formula that is logically equivalent to the original while still only using negation and conjunction)?

Answer to 3b

• For p|q, we start with the disjunction (pq) (qp) (pq) After getting rid of the disjunctions, this is much lengthier than the logically equivalent formula (pq).

(Lesson: the procedure in the proof does not always get you the shortest answer.)