Homework 1
due: January, 24, 2013
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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
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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.
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