December 1998
MA214: DISCRETE MATHEMATICS

QUESTION 1 (Compulsory)

Total Marks: 20 Marks

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SUGGESTED SOLUTIONS
for Question 1

 
(a) Suppose that A and B are the sets defined below.

A = {x | x is a real number such that x2 = 1}
B = {x | x is an integer such that x2 = 1}

 
(i) List the members of set A and write out in full the power set P(A). [2]
(ii) What is the cardinality of the power set P(B)? Show all of your working.

 

 [2]
(b) Let A be the vector matrix1.GIF (211 bytes)  and B be the vector  matrix2.gif (222 bytes). Calculate
(i) ABt [2]
(ii) AtB

 

 [2]
(c) Calculate the value of
formula1.gif (355 bytes)

 

 [2]
(d) Let the sets P, O and I be the sets described below.

P    The set of all prime numbers
O   The set of all odd numbers
I    The set of all integers

Taking the real numbers to be the universal set, draw a Venn Diagram showing the relationships between P, O and I.

 

 [3]
(e) Two classes, each containing 30 students, sit the same examination. The means and variances are given below.
  Mean Variance
Class A 67.3 5.9
Class B 73.4 18.2

Comment on the relative abilities of the two classes justifying your observations by referring to both the means and the variances.

 

 [3]
(f) The probability that an event A occurs is 0.6
The probability that an event B occurs given that event A has occurred is 0.8.
The probability that event B occurs given that event A has not occurred is 0.3

 
(i) Calculate the probability of events A and B both occurring. [1]
(ii) Calculate the probability of event B occurring and event A not occurring. [1]
(iii) Hence calculate the probability of event B occurring. [1]
(iv) Hence calculate the probability that event A occurred given that event B occurred. [1]