ADVANCED DIPLOMA IN COMPUTER STUDIES
PROGRESS TEST I
CHAPTERS 1 to 3
SECOND PROGRESS TEST
SET 1 MARKING SCHEME
MA214
MARKING SCHEME
NOT FOR STUDENTS
INSTRUCTIONS
1.
Do not allow students access to marking scheme under any circumstances.
2.
Mark all questions, using marking scheme as a guideline only.
3.
Take into account alternative answers.
Marking Scheme
Do not distribute to students
Question 1 (compulsory)
a)
i)
Find the power set of A = Φ [1 mark]
answer 1 mark for listing the power set
ii)
What is the cardinality of A and P(P(P(A)))? [2 marks]
answer
1 mark for A solution
2 mark for correct answer for P(P(P(A))))
b)
Let A = {2,3,4,5,6,7,8} defined by aRb if and only if 3 | (a-b).
Use counter-example to determine the following properties:
i)
ii)
iii)
reflexivity;
[2]
anti-symmetry;[2]
transitivity.
[2]
answer : each part 1 mark for justifying the solution 1 mark for correct identifying – no
mark for stating the answer
Prove, or disprove the given Boolean, clearly identify the laws used.
A ∪ (B ∩ C) ≡ (A ∪ B) ∩ (A ∪ C)
[3 marks]
1m for correct proving ;1m identifying the laws used.
d)
Let f(x) =2x and g(x) = 5x2 – 2x , find ( ( g o f) o f)(x)
[2 marks]
Answer 1 mark for find go f and 1 mark for having correct answer
e)
Write negations for each of the following statements:
a. John is less than six feet and he weighs more than 70 kg.
b. The bus was early or Tom’s watch is slow.
[2 marks]
1m for correct answer; max up to 2marks.
f)
Without using table, use the laws to show ( p ^q ) à ( p v q) is a tautology
[2 marks]
answer 1 mark for stating the law 1 mark for explicitly giving the steps to T
MARKING SCHEME
Marking Scheme
Do not distribute to students
Question 2 (Logic & Function)
a)
Determine the validity of the following using a truth table. [5 marks]
Oleg is a math major or Oleg is an economics major.
If Oleg is a math major, then Oleg is required to take Math 362.
∴Oleg is an economics major or Oleg is not required to take Math362
answer
1 mark for declaring the proposition
1 mark for defining the logical expression
Deduct 1 mark for 1 wrong entry in the truth table; up to a max 2 marks
1 mark for stating correct answer
b) Define f : z+ → z+ by the rule f(n) = 1+ 2n - n2 . Use arrow diagram to show that it
has the followings.
Is f a function?
Is f a 1 to 1 function?
Is f a onto function?
Justify your answer
[5marks]
No reason presented, no mark shall be awarded.
2 mark for identifying f is a function or not with reason
1 mark for giving a reason f is 1 to 1 function
1 mark for giving a reason f is a onto function
MARKING SCHEME
Marking Scheme
Do not distribute to students
Question 3 (Set theory)
1. Indicate which following relationships are true and which are false. Justify your
answer
i. Z ⊆ Q
ii. ((Z – )c ∩ N) ⊂ N
[2marks]
Answer 1 mark for part a) and 1 mark for part b)
2. UseVenn diagram to show that
(A – C) ∩ (C – B) = Φ
[2 marks]
deduct 1m for each error; up to max 2marks
3. Let U = set of lower letters, A = {a,b,c}, B= {b,c,d} and C = {b,c,e}
Find
i. A – (B U C)
ii. A ∩ (B – C)
[2 marks]
answer 1 mark for part I having correct answer 1 mark for part ii having correct answer.
4. Indicate whether the argument is valid or invalid. Support answers by drawing
Venn diagram
All discrete mathematics students can tell a valid argument from invalid one.
All thoughtful people can tell a valid argument from an invalid one.
∴All discrete mathematics students are thoughtful.
[4 marks]
answer deduct 1 m for each mistake made; up to a max 4
1m for two discs representing valid and invalid; the discs are not matual exclusive
1m for placing discrete mathematics in valid argument
1m for discrete in thoughtful
1m for a rectangle.
MARKING SCHEME
Marking Scheme
Do not distribute to students
Question 4 (Relations & Functions)
1. Let A = {1,2} and B = {1,2,3} and define a binary relation from R from A to B as
follows given any (x,y) ∈ A X B, (x,y) ∈ R ⇔ |x – y| is a prime number
a. State explicitly which ordered pairs are in A X B [1m]
b. Identify the how many functions are subset of A X B. [1m]
c. Is 1 R 2? Is 1 R 3? [2m]. Justify your answer.
1m correct listing the ordered pairs
1m 3*3 = 9 functions
1m 1 R 2 is not since 1-2 = 1, which is not a prime number
1m 1 R 3 is Yes since 1 –3 = 2, which is a prime number.
2. Let A = {2,4} and B = {6,8,10} and define binary relations R and S from A to B
as follows:
for all (x,y) ∈ A X B, xRy ⇔ x | y
for all (x,y) ∈ A X B xSy ⇔ y – 4= x
a. State explicitly which ordered pairs in RC ∪ S and R ∩ SC. [3m]
b. Determine whether R and S are functions. If yes, explain, whether R and S
are onto and1 to1 function. [3m]
1m for finding RC
1m for RC ∪ S
1m for R ∩ SC
Answer for R and S is 1m
Determine each property for R 1m
Determine each property for S 1m
Question 5 (Functions)
a. For the following functions f and g defined by the arrow diagrams below, find
g o f and f o g and determine whether g o f equals f o g .
[4 marks]
g o f is defined as follows:
( g o f )(1) = g ( f (1)) = g (5) = 1,
( g o f )(3) = g ( f (3)) = g (3) = 5,
( g o f )(5) = g ( f (5)) = g (1) = 3.
MARKING SCHEME
[1 mark]
Marking Scheme
Do not distribute to students
f o g is defined as follows:
( f o g )(1) = f ( g (1)) = f (3) = 3,
( f o g )(3) = f ( g (3)) = f (5) = 1,
( f o g )(5) = f ( g (5)) = f (1) = 5,
[1 mark]
Then g o f ≠ f o g because, for example ( g o f )(1) ≠ ( f o g )(1).
[2 marks]
b. Let X={1,4,5} and Y = { 1,3,4,5}. Define g: X à Y by the following arrow
diagram.
1
1
4
3
5
4
5
i. Write the domain of g and the co-domain of g? [2m]
ii. What is the range of g ? [1m]
iii. How many functions are there from a set of 3 elements to a set with 4
elements? [1m]
1m for domain g
1m for co-domain of g
1m for range of g
1 the functions.
c. Show that (f o f –1)(x) = (f -1 o f)(x) for f(x) = 2x. [2m]
1m for f –1
1m for proving both sides are equal
THE END
MARKING SCHEME