Exercises for CS1512 Weeks 7 and 8 Propositional Logic 1 (questions)

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: p q. Connectives: {, }. b.Formula: p q. Connectives: {,, }. c.Formula: p q. Connectives: {, }. d.Formula: (p q) (( p) q). Conn: {, }.

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).

Ex. 3. 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 p q.

Question 4a In class, it was proven that {, } is a functionally complete set of connectives. Making use of this result, can you prove that {, } is also functionally complete?

Question 4b In class, it was proven that {, } is a functionally complete set of connectives. Making use of this result, can you prove that {NAND} is also functionally complete? [Explanation: (p NAND q) is TRUE iff (p q) is FALSE. This connective is also called the Sheffer stroke and written (p|q).)

Question 4c Given this result, why do we bother defining and using more than one connective?

Question 5 Translate into propositional logic (abbreviating it has rained as r, its been cold as c, and the plant is dead as d): a. If it has rained and its been cold then the plant is dead b. If it has rained then either it hasnt been cold or the plant is dead Use truth tables to determine whether these two statements are logically equivalent