Faculty of Computer Studies
MT131 – Discrete Mathematics
Take Home Exam for Final Assignment (Fall 2020/2021)
Cut-Off Date: --Duration: 48 Hours
Total Marks: 100
Contents
Warnings and Declaration……………………………………………………………… 1
Question 1...……………………….…………………………………………………………… 2
Question 2…..………………………………………………………………………………….. 3
Question 3 ………….………..………………………………………………………………… 4
Question 4 ………….………..…………………………………………………………………. 5
Question 5 ………….………..…………………………………………………………………. 6
Plagiarism Warning:
As per AOU rules and regulations, all students are required to submit their
own FTHE work and avoid plagiarism. The AOU has implemented
sophisticated techniques for plagiarism detection. You must provide all
references in case you use and quote another person's work in your FTHE. You
will be penalized for any act of plagiarism as per the AOU's rules and
regulations.
Declaration of No Plagiarism by Student (to be signed and submitted by student
with FTHE work):
I hereby declare that this submitted FTHE work is a result of my own efforts
and I have not plagiarized any other person's work. I have provided all
references of information that I have used and quoted in my FTHE work.
Name of Student: ………………………………..
Signature: …………………………………………...
Date: ……………………………………………………
MT131 – Discrete Mathematics
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MT131 – Discrete Mathematics
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Answer the following questions:
Q‒1:
a) [4+6 marks] let
and be propositions. Simplify the following
compound propositions:
(
)) ((
)
i. (
).
(
) (
)) ((
) ).
ii. (
b) [4+6 marks] Let , and be any sets. Prove that
(
)
i.
.
(
) (
) (
).
ii.
MT131 – Discrete Mathematics
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Q‒2:
a) [3+3+4 marks] Consider the following street map:
You may travel only north or east.
i. What is the total number of different routes you may take from to
?
ii. What is the total number of different routes you may take from to
?
iii. Suppose that you are at a corner on a rectangular grid of streets. You
want to go to a corner blocks east and blocks north. What is the
total number of different routes you may take?
b) [5+5 marks] Three letters are chosen at random from the word
“calculator". What is the probability that
i. One of them is a “t”?
ii. One or more of them is a “c”?
MT131 – Discrete Mathematics
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Q‒3:
*(
):
a) [3+3+2+2 marks] Let
1
2+ be a relation on
the set
*1 2 3 4+.
i. List the members of .
ii. Write the matrix representation of .
iii. Determine whether is symmetric or transitive.
iv. Find the symmetric and reflexive closures of .
(
)
b) [10 marks] Define the relation on as
iff
such that
2 . Show that is an equivalence relation.
MT131 – Discrete Mathematics
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Q‒4:
a) [5 marks] Let
be the partial order relation defined on
*1 2 3 4 6 8 12+, where
means
. Draw the Hasse diagram for .
( ) 2
b) [5 marks] Let :
1. Show that is both injective
and surjective.
c) [5 marks] Find the base-8 representation of 100 and base-9
representation of 1000. Show details of your work.
d)[5 marks] Find all
1 such that 27 9 (mo
).
MT131 – Discrete Mathematics
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Q‒5:
a) [10 marks] Find an adjacency matrix for .
b) [10 marks] Draw the undirected graph with adjacency matrix:
1 3
4
1 2 1 3
3 1 1
1
3
2
[4
1 2 3]
MT131 – Discrete Mathematics
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