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PC TRAINING & BUSINESS COLLEGE (PTY) LTD
HIGHER EDUCATION AND TRAINING
FACULTY OF MEDIA, INFORMATION & COMMUNICATION TECHNOLOGY
BSc INFORMATION TECHNOLOGY
MATHEMATICS 611 (MEMO)
1ST SEMESTER NATIONAL EXAM MEMO
DURATION: 2 HOURS
MARKS: 100
DATE: JUNE 2016
EXAMINERS:PHUMLANI SHABALALA
MODERATOR: EMMANUAL MANY
This paper consists of 5 questions and 5 pages including this page.
PLEASE NOTE THE FOLLOWING:
1) Ensure that you are writing the correct CA TEST 1paper, and that there are no missing pages.
2) You are obliged to enter your learner details on the answer booklet. The answer booklets provided are
the property of the PC Training & Business College and all extra booklets must be handed to the invigilator
before you leave the examination room.
3) If you are found copying or if there are any documents / study material in your possession, or writing on
parts of your body, tissue, pencil case, desk etc., your answer booklet will be taken away from you and
endorsed accordingly. Appropriate disciplinary measures will be taken against you for violating the code of
conduct of PC Training & Business College Examinations Board, therefore if any of these materials are in
your possession you are requested to hand these over to the invigilator before the official commencement
of this paper.
4) The question paper consists of 3 sections.
a. Sections A and B are compulsory.
b. Section C comprises of 3 questions, you are required to answer any 2 questions.
NUMBERS
SUGGESTED TIME REQUIRED TO ANSWER THIS QUESTION PAPER
QUESTIONS
MARKS
TIME IN MINUTES
SECTION A: MULTIPLE CHOICE QUESTIONS COMPULSORY
1
2
3
4
5
Question One
30
20
SECTION B: SHORT QUESTIONS COMPULSORY
Question Two
40
SECTION C: ANSWER ANY TWO QUESTIONS
Question Three
30
Question Four
30
Question Five
30
TOTAL
100
40
30
30
30
120
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SECTION A: MULTIPLE CHOICE QUESTIONS
QUESTION ONE
Four alternatives are provided for each of the following questions. Choose the correct alterna
questions/statements. Write down the question number in your answer book and the alphabet correspondin
the question number.
1.1
1.2
In a truth table for a two-variable argument, the first guide column has the following truth values:
A.
T,T,F,F
B.
C.
D.
T,F.T,F
F,T,F,T
T,T,T,T
The expression a + a̅ cis equivalent to ……………….
A.
a̅
B.
a+c
C.
D.
1.3
In propositional logic which one of the following is equivalent to p → q?
A.
p̅ → q
B.
p → q̅
C.
p̅ ∨ q
D.
1.4
c
1
p̅ ∨ q̅
̅BC + ABC
̅̅̅̅+ ̅̅̅̅̅̅
The minimized expression of ABC̅+ A
ABC is ………..
A.
A +C̅
B.
̅C
B
C̅
C.
D.
1.5
C
Let p be “He is tall” and let q “He is handsome”. Then the statement “It is false that he is short or
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handsome” is …………………..
A.
p∨q
B.
~ (~ p ∨ q)
C.
p∨~q
D.
p∧ ~q
1.6
Identify the converse of the following assertion:
I stay only if you go.
A.
I stay if you go.
B.
If you do not go then I do not stay
C.
D.
1.7
If I stay then you go.
If you do not stay then you go.
The contra-positive of given statement “If it is raining, I will take an umbrella” is …
A.
I will not take an umbrella if it is not raining.
B.
C.
D.
I will take an umbrella if it is raining.
It is not raining or I will take an umbrella.
It is sunny and I will not take an umbrella
1.8
A statement is also referred to as a …………….
A.
conclusion
B.
fact
C.
order
D.
proposition
1.9
In a disjunction , even if one of the statement is false the whole disjunction is still
A.
false
B.
negated
C.
true
D.
1.10
true or false
the bitwise XOR for a bit strings 0100010100 and 1010110111 is
A.
1110110111
B.
1110100011
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C.
D.
0000010100
none of above
1.11 If X and Y are two sets, then X ∩ (X U Y)cequals
A. x
B. y
C. Ф
D. none the above
1.12. A bi-conditional statement, even if one of the statement is false the whole proposition becomes
A. true
B.true or false
C. false
D. all of the above
1. 13.Let A ={ 1,2,3} and B ={ 1,2,3,4}. The relations R1 ={ (1,1), (2,2), (3,3)} andR 2 ={ (1,1), (1,2),(1,3),(1,4)}.
Find R1 – R2
A. {(2, 2), (3, 3)}
B. { (1,2) (1,3) (1,4)}
C. {(1,1), (2,2), (3,3)}
D. none of the above
1.14The domain and range of the function f(x) = √x - 2is
A. All real numbers and all positive numbers
B. All real number smaller than or equal to 2 and all positive numbers
C.All real number greater than or equal to 2 and all positive numbers
D.All real number excluding 2 and all positive numbers
1.15For the sequence an = 6. (1/3)n, a4 is ______.
A. 2/25
B. 2/27
C. 2/19
D. 2/13
1.16 Let A = { -5, 2} and B = {1, 0} . Find 2A + 3B.
A.(-10 , 4 )
B. (-8 , 6)
C. ( -9 , 4)
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D.(-7 , 4)
1.17Let R be the set of real numbers. If f: R -> R is a function defined by f(x) = x2 then f is
A. injective but not subjective
B. subjective but not injective
C. bijective
D. none of the above
1.18 If f(x) = 3x -5, then f-1
A. is given by 1/3y-5
B. is given by y + 5/ 3
C. does not exist because is not a one-to-one and not onto
D. does not exist is because is not invertible
no answer (students shall be given one mark)
1.19The function f: R -> R defined by f(x) = sin x is
A. into function
B. one-two-one function
C. onto function
D. many-one function
1.20. Bit string for the set {1,3,5,7,9} (with universal set {1,2,3,4, 5,6,7,8,9,10}) is10 1010 1010 .
What is the bit string for the complement of this set?
a. 1010101010
b. 1010101011
c. 0101010101
d. None of the above
1.21The Fibonacci sequence, f0,f1,f2,...,is defined by the initial conditions f0 =0,f1 =1, and the recurrence relationfn =
fn−1+fn−2 for n =2,3,4,....
The value of f5 is equally to ?
a. 3
b. 5
c. 7
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d. 9
1.22. Which one is the example of a disjoint sets?
A. A ={1,3,5,7,9} and B = {2,4,6,9,10 }
B. A ={1,3,5,7,9} and B = {2,3,6,8,10}
C. A ={2,3,5,7,9} and B = {2,4,6,8,10
D. A ={1,3,5,7,9} and B = {2,4,6,8,10}
1.23in the graph G , G = (V, E) the symbols V and E represent
A. Directed and undirected
B. Node and vertices
C. Edges and links
D. Vertices and Edges
1.24The relation R {(1,1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 2), (1, 3), (3, 1)} on set A = {1, 2, 3} is an equivalence relation since it
A. symmetric
B. transitive
C. reflexive
D. all of the above
1.25The total number of vertices adjacent to the vertex is called
A. adjacent vertex
B. neighborhood
C. degree of vertex
D. path of vertex
1.26. If a set A contains three elements, then its power set contains …………… elements.
A. 6
B. 8
C. 7
D. 9
1.27 An unordered collection of objects is called...
A. function
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B. set
C. graph
D. table
1.28The intersection of the sets {1,3,5} and {1,2,3} is
A. {1,2}
B. {1,2,3,5}
C. {1,3}
D. none of the above
1.29let f(x) = x2 and g(x) = √x , then f(g(x)) is
A. x2
B. x
C. x2+ √x
D. x1/2
1.30In graph a path is defined as a
A. is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.
B. is a sequence of edges that begins at point zero of a graph and travels through vertexes.
C. a number of edges in a graph
D. a size of the graph
SECTION B (COMPULSORY)
QUESTION TWO
2.1
Briefly explain the difference between propositional logic and predicate calculus, And specify the symb
that are used in both propositional logic and predicate?
Any valid explanation from students.
2.2
Verify that the functions f and g are inverse to each other
2.2.1
1
3
f(x) = 2 x3 - 2 and g(x) √2x + 4
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3
(g(x)) = 1/2( √2x + 4 )3 - 2
= 1/2 (2x + 4) - 2
=x+2-2
= x
3
g(f(x) = √2(1/2 x3 -2) + 4
3
= √x 3 -4
= x
+ 4
Since f(g(x)) is equal to g(f(x)) function f and g are inverse to each other.
2.3
Find f(g(x)) for the following functions.
1
2.3.1
f(x) = 2x - 4 and g(x) = x + 2
2
=> f(g(x)) = 2(1/2 x + 2) - 4
= x +4-4
= x
2.4
Use quantifiers and connectives to translate the following English statements into predicates.
2.4.1Everyone at Richfield is smart
ᵾx atRichfield(x) -> Smart(x)
2.4.2Some dogs are useless
∃x Dogs(x) ^ Useless(x)
2.5
Use identities and logical equivalence to show that~ (p ν (~p ᴧ q)) = ~p ᴧ ~ q.
LHS
~(p v (~p ^ q))
(~p ^ ~ (~p ^ q))
Morgans Law
(~p ^ (p ^ ~q))
Double Negation
(~p ^ p) v (~p ^ ~q)
Distributive law
F v (~p ^ ~q)
Negation Law
(~p ^ ~ q)
Proven
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Build a digital circuit that produce the output ((~p∨ ~r)∧~q)∨ (~p∧ (q∨ r)) when given input bits p, q a
Digital circuit is attached
2.6
SECTION C (ANSWER ANY TWO QUESTIONS FROM THIS SECTION)
QUESTION THREE
3.1
Let’s Q(x, y) be the statement 2x - y = 0. The domain for both x and y is the set of inte
What is the truth values of the following?
3.1.1 Ǝx Q(x,4)
2x - 4 = 0
2(2) - 4 = 0
0 = 0 true (there exist integer 2)
3.1.2 Ʉy Q(1,y)
2x - y = 0
2(1) - 3 = 0
-3 = 0
false (3 is an integer as well)
3.2
Construct circuits that produce the following outputs:
Circuits are attached
̅̅̅̅̅̅̅̅
3.2.2
x̅(y
+ z̅ )
3.2.3
(x + y + z)(x.
̅ y̅.z̅ )
3.3
Find the product of the following matrices, show your work ?
0 -4
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1 -1
-4 - 4
3.4
Find the value of the following summation, show your work?
Methods one
3j + 2i
0
1
2
2j
0
2.0 + 3.0 = 0
2.1 + 3.0 = 2
2.2 + 3.0 = 4
6
3i
1
2.0 + 3.1 = 3
2.1 + 3.1 = 5
2.2 + 3.1 = 7
15
2
2.0 + 3.2 = 6
2.1 + 3.2 =8
2.2 + 3.2 = 10
24
3
2.0 + 3.3 = 9
2.1 + 3.3 = 11
2.2 + 3.3 = 13
33
Total
Methods two
=> ∑3I= 0(3i + 2.0) + (3i + 2.1) + (3i + 2.2)
9i + 6
9.0 + 6) + (9.1 + 6)+ (9.2 + 6) + (9.3 + 6)
= 78
3.5 Given set A = {1,2,5,8,9} , B = {5,6,7,8,9}, C {1,2,5,8,9, 10}, D = {10,11}
Find (A∩B ) ν (C∩ D) ?
(A ^ B) = {5, 8, 9}
(C ^ D) = {10}
(A∩B ) ν (C ∩ D) = {5, 8 , 9, 10}
QUESTION FOUR
4.1
Prove by contradiction that √2 is irrational?
show all your work
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78
 Suppose √2 is rational,
 There must be two integers a and b such that √2 = a/b
Assume that a and b have no common factor
√2 = a/b
√2 (b) = a/b (b)
(multiply both side by b)
√2 (b) = ab/b
√2 (b) = a
2b^2 = a^2
( square on both side )
this shows that a^2 is even
if a^2 is even then a is even
=> ****a = 2k
(where k is any integer)
=> 2b^2 = a^2 = (2k)^2 = 4K^2
=> 2b^2 = 4k^2
so b^2 = 2k^2
=> b^2 is even
=> **** b = 2k
(where k is any integer)
Since a and b are both even and so are multiples of 2
This is a contradicts a and b should not have a common factor , thus
√2 is irrational
4.2
Use a Truth table to verify (p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r).
Truth table
p
T
T
T
T
F
F
F
F
4.3
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
qvr
T
T
T
F
T
T
T
F
p ^ (q v r)
T
T
T
F
F
F
F
F
p^q
T
T
F
F
F
F
F
F
p^r
T
T
T
F
F
F
F
F
(p^q)v(p^r)
T
T
T
F
F
F
F
F
Duality principle state that every algebraic expression deducible from the postulates of Boolean algebra
remains valid if the operations and identity elements are interchanged
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Find the duals of the following Boolean expression.
4.3.1 x(y + 0 + Z)
x+y.1.z
4.3.2 XYZ
x+y+z
4.4
Construct half adder that can perform addition of two inputs a and b
Half Adder is attached
QUESTION FIVE
5.1 Briefly define the term bipartite graph, use example to support your answer?
Direct definition from student's guide, example to support.
5.2 Represent the graph shown below with the incidence matrices
Incidence Matrix
v1
v2
v3
v4
v5
e1
1
0
0
0
0
e2
1
1
0
0
0
e3
1
1
0
0
0
e4
0
1
1
0
0
e5
0
0
1
0
1
e6
0
1
0
0
1
e7
0
1
0
1
0
Find the degree in and out and neighborhoods of the graph below
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e8
0
0
0
1
0
(10)
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5.3
deg-(a) = 3
deg-(b) = 2
deg-(c) = 3
deg-(d) = 2
deg-(e) = 3
deg-(f) = 0
deg+(a) = 4
deg+(b) = 1
deg+(c) = 2
deg+(d) = 2
deg+(e) = 3
deg+(f) = 0
N(a) = {b,c,e}
N(b) = {a,c,d}
N(c) = {a,b,d}
N(d) = {c,b,e}
N(e) = {a,d}
N(f) = {}
5.4 Direct definition from student's guide.
TOTAL MARKS: 100
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