Department of Systems Engineering and Engineering Management
The Chinese University of Hong Kong
LINEAR ALGEBRA & DISCRETE MATHEMATICS
SEG2410A/B
Exercise 2
Question 1.
State whether each of the following is true or false. In each case give reasons for your
answer.

(a) n  1  On 2  ,

(b) 2 2 n  O 2 n .
2
 
Question 2.
Let A be a positive constant. Show that An < n! for sufficiently large n.
Hint: Analyze Example 3(b) of the Textbook “Ross and Wright” (P. 46).
Question 3.
Convert each of the following arguments into logical notation using the suggested
variables. Then provide a formal proof.
“If my computations are correct and I pay the electric bill, then I will run out of
money. If I don’t pay the electric bill, the power will be turned off. Therefore, if I
don’t run out of money and the power is still on, then my computations are
incorrect.” (c, b, r, p)
Question 4.
Define the exclusive OR connective p  q as
 p  q 

Using the truth tables, prove that

(a)  p  p  is a tautology,

(b)  p  p  p 

(c) p  q   p  q .
p,
p  q .
Question 5.
Consider the table of the rules of inference. For example, the “modus tollens” rule of
interence.
PQ  Q logically implies P
A “real-life” example of this rule would be


P = “Mathematicans”.
Q = “Poor”.
The argument goes like this: All mathematicans are poor. But John is not poor.
Therefor, John is not a mathematican.
If we have the hypothesis
-
A  P , and
A B,
give a proper list of inference for reaching the conclusion P   B .
Question 6.
Suppose n 3,4,5,6,, consider the following statements



(1) If there exist k    such that n  3k , then the term n 2  2 is never
divisible by 3.
(2) If there exist k    such that n  3k  1 , then the term n 2  2 is never
divisible by 3.
(3) If there exist k    such that n  3k  2 , then the term n 2  2 is never
divisible by 3.
To prove statement (1), we adopt a contradiction proof as follows:
Suppose there exist k    such that n  3k , and n 2  2 is divisible by 3. Then, there
exists p    such that n 2  2  3 p . Thus, we have
3k 2  2  3 p ,
or


3 3k 2  p  2 .
Let q  (3k 2  p) . Since both k and p are positive integers, thus q is also an integer.
However, we have
q
2
,
3
which shows that q is not an integer. Hence, this gives a contradiction!!!
Therefore, n 2  2 is never divisible by 3 for all n expressible as n  3k .

(a) Prove statements (2) and (3) by contradictions.

(b) Using the results from statements (1), (2) and (3), prove the following
statement
“The term n 2  2 is never divisible by 3 for all n 3,4,5,6,.”

(c) What type of proof are we using in (b)?
Question 7.
Substitution Rules:

(1) If a compound proposition P is a tautology and if all occurrences of some
variable of P, say p, are replaced by the same proposition E, then the resulting
compound proposition P* is also a tautology.
For example, recall that  p   p  q  q is a tautology. Replace q by q  p ,
we have  p   p  q  p q  p , which it can be shown that it is a
tautology.

(2) If a compound proposition P contains a proposition Q, and if Q is replaced by a
logically equivalent proposition Q*, then the resulting compound proposition P* is
logically equivalent to P.
For example, let P be the compound proposition  p  q    p  q  . From De
Morgan’s Law, we know that  p  q   p  q  . Hence, by replacing
 p  q in P with  p  q  , we have P*, the new compound proposition,
  p  q     p  q . It can be shown that P* is logically equivalent to P.
Simplify the following statements without using the truth table


(a) q   p  q  r      p  q   p  r  ,
(b)  q  r    p    p  r    q.