April 1999
MA214: DISCRETE MATHEMATICS

QUESTION 4

Total Marks: 20 Marks

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GRADE A
Student's solutions are indicated in green.
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(a) Consider the statement S:

S = "if p is a prime number then p is odd"

 
(i) Write down the inverse of S. [1]
(ii) Write down the converse of S. [1]
(iii) Find counter examples to provide that none of these three statements are tautologies. [2]
(i) If p is not a prime number then p is even.

(ii) If p is odd then p is a prime number.

(iii) 3 counter examples:

  1. For S
    If p is a prime number then p is odd.
    Example: 2 is a prime number but 2 is not odd.
  2. For inverse of S
    If p is not a prime number then p is even.
    Example: 1, 9, 15 are not prime number, they are not even too.
  3. For converse of S
    If p is odd then p is a prime number.
    Example: 9 is odd but 9 is not a prime number.

 
(b) The fourth term in a geometric sequence is -72, and the seventh term in the sequence is +9.
(i) Calculate the common ratio. [2]
(ii) Write down the first three terms of the sequence. [1]
(iii) Calculate the sum to infinity of the sequence. [1]
(i) arn-1 = value of that term
ar3 = -72 ®  (1)
ar6 = 9 ®  (2)
Let a = -72/r3
Substitute a into (2)
-72 * (r6/r3) = 9
r3 = -9/72
    = -1/8
  r = -1/2
Substitute r = -1/2 into (1)
-72 = (-1/2)3 * (a)
-72 = -a/8
  a = 72 * 8
     = 576
Common ratio r = -1/2

(ii) The first 3 terms = a, ar, ar2
                               = 576, -288, 144

(iii) Since |r| < 1, therefore
S¥ = a / (1 - r)
      = 576 / (1 + 1/2)
      = (576 * 2) / 3
      = 384

 
(c) Construct the truth table for (~p « ~q) « (q « r) [4]
 
p q r ~p « ~q q « r (~p « ~q)  « (q « r)
T T T T T T
T T F T F F
T F T F F T
T F F F T F
F T T F T F
F T F F F T
F F T T F F
F F F T T T

 
(d) What can you say about the sets A and B if the following are true?
(i) A È B = A [1]
(ii) A Ç B = A [1]
(iii) A - B = A [1]
(iv) A - B = B - A [1]
(i) A È B = A
B is a subset of A.

(ii) A Ç B = A
A is a subset of B.

(iii) A Ç B = Æ
They are mutually exclusive.

(iv) A = B
Both set A and B have identical elements in their sets.

 
(e) Use algebraic laws show that pÙ(~pÚq) º pÙq [4]
p Ù (~p Ú q) º p Ù q
LHS = p Ù (~p Ú q)
       = p Ù ~ p Ú p Ù q (distributive law)
       = Æ Ú p Ù q (Complement law)
       = p Ù q (Identity law)