MTH 200 – Graded Assignment 2
Material from Module 3 (Logic and Proof) and Module 4 (Operations and Operators)
P1: Let p= "birds fly", q = "dogs howl", r = "elephants are pink".
a) Translate the compound statement into symbols using the logical operators , ,  :
It is not the case that birds do not fly and dogs howl, or elephants are pink.
b) Translate into English: p  (r  q)
P2: Construct a truth table for the expression
q  ( p  r )
p
q
r
q
pr
q  ( p  r )
P3: Write the inverse, converse and contrapositive of the statement
" 1  1  4 " implies " 2  5  7 "
What are the truth values of the original statement and its rearrangements?
P4: Determine whether the arguments are valid or invalid and cite the form of the syllogism /
fallacy (show translation to symbolic form):
a) Either the sun is shining or the sky is blue.
The sun is not shining.
The sky is blue.
b) If my candidate wins the election, your taxes will be lower.
Your taxes were lowered by the winning candidate.
My candidate must have won.
P5: Prove that the product of two odd numbers is odd. Use a direct proof.
P6: Prove that if n is an integer and 3n  5 is odd, then n is even using
a) an indirect proof
b) a proof by contradiction
P7: Which of the properties (closure, associative, identity, inverse, commutative) are satisfied by
 A,*  , where A  {a, b, c} and * is defined by the Cayley table given below? Which
properties fail? Explain – each property must have some justification/explanation as to what
you’re looking at that supports your answer.
P8: Which of the properties (closure, associative, identity, inverse, commutative) are satisfied
for the logical operation of  (“or”)? Explain – each property must have some
justification/explanation as to what you’re looking at that supports your answer. Hint: you
might find it helpful to start by turning the truth table for p  q into a Cayley table for the 
operator on the set {T , F } .
P9: Which of the properties (closure, associative, identity, inverse, commutative) are satisfied
for the set operation of  defined on the set of subsets of some universal set? Prove each
property which holds, and give a counterexample or explanation for the ones that fail.