Semester Two Final Examinations, 2019
MATH7861 Discrete Mathematics
This exam paper must not be removed from the venue
Venue
Seat Number Student Number Family Name First Name
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School of Mathematics & Physics EXAMINATION
Semester Two Final Examinations, 2019
MATH7861 Discrete Mathematics
This paper is for St Lucia Campus students.
Examination Duration:
Reading Time:
Exam Conditions:
This is a Central Examination
This is a Closed Book Examination – no materials permitted During reading time – write only on the rough paper provided This examination paper will be released to the Library Materials Permitted In The Exam Venue:
(No electronic aids are permitted e.g. laptops, phones)
Calculators – Casio FX82 series or UQ approved (labelled)
Materials To Be Supplied To Students:
None
Instructions To Students:
Additional exam materials (eg. answer booklets, rough paper) will be provided upon request.
Answer all questions in the space provided. Show all working; answers given without justification may not receive full marks. Questions carry the number of marks shown.
Total marks available: 70
120 minutes 10 minutes
For Examiner Use Only
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Semester Two Final Examination, 2019 MATH7861 Discrete Mathematics
Answer each question in the space provided on this examination paper, using the backs of pages if more space is required. Each question is worth the number of marks indicated on the right.
1. For statement variables p, q and r, prove that (⇠(q!⇠p)^r)_(⇠(p!q)^r) ⌘ p^r.
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(3 marks)
Semester Two Final Examination, 2019 MATH7861 Discrete Mathematics
2. Leta=p2q7r4 andletb=p3q4swherep,q,randsareprimeswithp
distinct prime numbers p ,p ,…,p , and positive integers e ,e ,…,e such that n = pe1pe2 ···pek, 12k 12k 12k
and any other expression for n as a product of prime numbers is identical to this except perhaps for the order in which the factors are written.
Schro ̈der-Bernstein Theorem For all sets A and B, if |A| |B| and |A| |B| then |A| = |B|. Binomial Theorem Given any real numbers a and b, and any nonnegative integer n,
(a+b)n =Xn ✓nk◆an kbk. k=0
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(b)A\Ac =; (b)A\A=A
(b) A \ ; = ;
A B = A \ Bc
The Quotient-Remainder Theorem Given any integer n and any positive integer d, there exist