# MA161 Assignment 1 202001 F2F Final

```DISCRETE MATHEMATICS I
MA161 Assignment 1
Semester II, 2020
TOTAL: 60 marks
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DUE DATE: WEEK 6 Sunday 11.59 pm (FJT)
13 / 09 / 2020
NOTE: 1. All questions are compulsory.
2. Show all relevant working.
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Instructions:
1. This assignment is due on Sunday, 13th September, at 11.59:00 PM Fiji Time. Upload it on Moodle.
2. There are FOUR questions. Answer all of them. Your solutions must be in the same chronological order
as the given questions.
3. To obtain full marks on a question, you have to show all your working, even when the question does not
say so.
4. You are to TYPE your solutions and submit them as a PDF document (single document) on Moodle.
5. Remember to click the SAVE CHANGES button when you upload your solutions.
6. This is a group assignment (maximum of 3 people in a group)! You are not allowed to copy from others,
or let others copy from you. If copying has been detected, everyone involved will forfeit all marks.
Plagiarism of any form is not accepted. Individual submissions will also be accepted.
Question 1
(i). Write the truth table for the proposition   r  q    p  r  .
(ii). Determine whether  p  q    p  q   q .
(iii).Write the converse, inverse, contrapositive and negation of the statement “You can check out library
books if you are registered.”
[3 + 3 + 4 = 10 marks]
Question 2
Let F  A  be the predicate “ A is a finite set” and S  A, B  be the predicate “ A is contained in B ”. Suppose
the universe of discourse consists of all sets. Translate the statement into symbols.
(i). Not all sets are finite.
(ii). Every subset of a finite set is finite.
(iii).No infinite set is contained in a finite set.
(iv).The empty set is a subset of every finite set.
[2 + 3 + 3 + 2 = 10 marks]
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Question 3
(i). Prove that if x and y are real numbers such that x  y, then x 
x y
 y.
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(ii). Prove or disprove that “Every integer is less than its cube.”
(iii).Prove or disprove that max  x, y  
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 x  y  x  y  , where x, y  .
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(iv).Prove or disprove that x  x , for x  .
(v). Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if
3n  5 is even, then n is odd. Which type of proof would be easier to give? Explain why.
[5 + 2 + 5 + 3 + 5 = 20 marks]
Question 4
(i). Show that if A, B and C are sets, then
A B C  A  B  C  A B  AC  B C  A B C .
(ii). Prove or disprove that for all sets A, B and C , we have
a)
A  B  C    A B   A C 
b) A   B  C    A  B    A  C 
(iii).Show using the membership table that A  B  C  A  B  C .
[5 + 5 + 5 + 5 = 20 marks]
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