Assignment 1
COMP2130 Winter 2014.
Due Jan. 31, 2014
Obey all instructions given in the ROASS document.
Q1) Construct a truth table and a justifying sentence to show that the statement 1 and statement 2 are logically
equivalent or not.
a) 1) pt 2) t
b) 1) ~(p q) 2) ~p ^ ~q
Q2) Use De Morgan’s Laws to write the negation for the following statements
a) John is a brown belt and Val is a blue belt.
b) 0 < x  9.
Q3) Use truth tables to establish whether the following statement forms a tautology or a contradiction or neither.
((q^r)^(~p^q)) ^ ~q
Q4) If in question 3 I had left out the words or neither, what is the minimum number of lines of the truth table
necessary to answer the question and why. I want some logic here!
Q5) Assume x is a particular real number. Determine whether statement (a) and (b) are logically equivalent
a) x < 2 or it is not the case that 0 < x < 3.
b) x  0 or either x < 2 or x ≥ 3.
Do this question by letting p,q,r be the 3 statement variables for the 3 statements in a) and b). The write out a)
and b) in symbols and now construct a truth table and a justifying sentence.
Q6) Use the logical equivalences on Page 35 in Theorem 2.1.2 to prove the logical equivalence of the
following. Show all steps. Give the law being used in each step.
~(p (q^ ~p))  ~p ^ ~q
Q7) Construct the truth table for (pq)  (q  p) and pq. What can you conclude?
Q8) Show that the following statement forms are both logically equivalent using truth tables.
a) p  q  r b) p ^ ~q  r
Q9) Write the contrapositive, converse and inverse of the statement “If P is a square then P is a rectangle.”
Are all the statements true?
Q10) Write the following statement, “It is sufficient for John to win this frame, to win the bowling trophy” in
the usual “if p then q” style.3
Q11) Write a logically equivalent statement form for “(qp) ^ (~p q)” without using “” or “”. Then
find a logically equivalent statement to your answer that is as simple as possible. Back up your statements with
authorities .
Q12) Use truth tables to determine which of the following arguments are valid or not. Check off the critical
rows and give the reason for you answer.
a) 1) p
2) q
p ^ q
b) 1) pq
2) qr
 pr
Q13) Use symbols to write the logical form of the following argument. If the argument is valid, identify the
rules inference that guarantees its validity. Otherwise, state whether the converse or inverse error was made.
If at least one of two numbers is divisible by 6 then the product of the two numbers is divisible by 6. Neither of
the two numbers is divisible by 6. Therefore, the product of the two numbers is not divisible by 6.
Q14) You are visiting the island of knaves and knights where knaves always lie and knights never lie. You
meet two natives of the island. A says “B is a knave.” B says “A is a knave.” How many knaves are there?
Prove it. You can do it like in class or use more formal methods
Q15) Use the valid argument forms listed in Table 2.3.1 on page 61 to deduce the conclusion from the premises.
a) ~r ~q
b) p ^ s  t
c) ~r
d) ~q u ^ s
e) p  q
f) t