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Homework 1 due: January, 24, 2013 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) Table of equivalences p∧q q∧p p∨q q∨p ¬(¬p) p (p ∨ q) ∧ r (p ∧ r) ∨ (q ∧ r) (p ∧ q) ∨ r (p ∨ r) ∧ (q ∨ r) (p ∨ q) ∨ r p ∨ (q ∨ r) (p ∧ q) ∧ r p ∧ (q ∧ r) p⇒q ¬p ∨ q p⇔q (p ⇒ q) ∧ (q ⇒ p) ¬(p ∧ q) ¬p ∨ ¬q ¬(p ∨ q) ¬p ∧ ¬q p∧T p p∨F p p∧F F p∨T T p∧p p p∨p p p ∨ ¬p T p ∧ ¬p F p ∨ (p ∧ q) p p ∧ (p ∨ q) p Problems marked with ∗ will be graded. Please practice on the rest of them to the extent you find useful. 1. ∗ Construct truth table for ¬p ∧ ¬q, as well as for ¬(p ∨ q). Are those statements equivalent? 2. Construct truth table for ¬p ∨ ¬q, as well as for ¬(p ∧ q). Are those statements equivalent? 3. ∗ Construct truth table for (p ∨ q) ∧ r and for (p ∧ r) ∨ (q ∧ r). Are those statements equivalent? There are three variables here, so your table should have 8 rows. 4. Construct truth table for (p ∧ q) ∨ r and for (p ∨ r) ∧ (q ∨ r). Are those statements equivalent? There are three variables here, so your table should have 8 rows. 5. ∗ Using the table of equivalences, prove that p ∨ q is equivalent to ¬p ⇒ q, and that it is also equivalent to ¬q ⇒ p 6. ∗ Using the table of equivalences, prove that (p ⇒ r) ∨ (q ⇒ r) is equivalent to (p ∧ q) ⇒ r 7. Using the table of equivalences, prove that (p ⇒ q) ∨ (p ⇒ r) is equivalent to p ⇒ (q ∨ r) 8. In the problems below, determine if the two statements are equivalent. If they are, prove it using the table of equivalences, if they are not, find a counterexample. (a) ∗ p ⇔ q and (p ∧ q) ∨ (¬p ∧ ¬q) (b) ∗ (p ⇒ q) ⇒ r and p ⇒ (q ⇒ r) (c) ¬(p ⇔ q) and p ⇔ ¬q (d) ¬p ∧ ¬(p ∨ q) and ¬p ∨ q 1 Remarks: • If the task is to construct a truth table, only the final product, the table, will be graded. Your scratch calculations, to figure out whether the entry should be T or F , does not have to be in your answers. • In the problems for which you are to prove something, please supply a sequence of propositions starting from LHS and ending with RHS, separated by ’≡’ sign, with the number of entry from the table of equivalences above ’≡’ sign, you are using in a given step, just like during the lecture. • In a problem where you decide that two statements are not equivalent and you want to present a counterexample, it is enough to state what values of p, q and r you choose and what values two statements, judged non equivalent, will have. 2