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MATH1061 Discrete Mathematics
创建日期：2024-04-19 15:54:03
所属分类：数学mathematics
标签： MATH1061
Discrete Mathematics

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MATH1061 Discrete Mathematics

Page 1 of 15
This exam paper must not be removed from the venue
School of Mathematics & Physics
EXAMINATION
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.
Complete all examination questions in the space provided on this examination
paper. If more space is required, use the backs of pages. The final page is a
formula sheet, which you may detach. Nothing written on the formula sheet will
be marked.

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
∼ ((p→ q) ∧ (p→ r)) ≡ p ∧ (∼ q ∨ ∼ r).
(4 marks)
Page 2 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
2. Consider the sequence defined recursively as
a1 = 2, a2 = 3, a3 = 4 and ak = ak−1 + ak−2 + ak−3 for all integers k ≥ 4.
Prove that an ≤ 2n for each integer n ≥ 1.
(6 marks)
Page 3 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
3. Indicate whether each of the following statements is true or false, by circling True or False as
appropriate. No explanation is required.
(5 marks)
(a) Z ∈ Q.
True False
(b) ∀ sets A, ∅ ∈ A.
True False
(c) If A = {∅} then |P(A)| = 2.
True False
(d) ∀ sets A and B, if |A| = |B| then every one-one function f : A→ B is an onto function.
True False
(e) If A = {a, b, c} and B = {d, e} then f = {(a, d), (b, d), (c, d)} is a function from A to B.
True False
(f) Let A be a non-empty set, and let ρ be a relation defined on P(A) where ∀X, Y ∈ P(A),
XρY if and only if X ∩ Y 6= ∅.
(i) ρ is reflexive.
True False
(ii) ρ is symmetric.
True False
(iii) ρ is transitive.
True False
(g) (Z+,×) is a group, where × represents multiplication.
True False
(h) Let A = {1, 2, 3, 4}. The set of all functions from A to A with the binary operation of
composition of functions is a group.
True False
Page 4 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
4. Prove the following statement. For all sets A and B, P(A) ⊆ P(B) if and only if A ⊆ B.
(5 marks)
Page 5 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
5. Prove that the set Z+ × Z+ × Z+ × Z+ is countable.
(7 marks)
Page 6 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
6. For each positive integer x, define d(x) to be the number of positive divisors of x. Let ρ be the
relation on Z+ defined as follows: ∀m,n ∈ Z+, (m,n) ∈ ρ if and only if d(m) ≤ d(n).
(a) Explain why (1, n) ∈ ρ for all n ∈ Z+.
(1 mark)
(b) Prove or disprove the following statement: the relation ρ is reflexive.
(2 marks)
(c) Prove or disprove the following statement: the relation ρ is symmetric.
(2 marks)
(d) Prove or disprove the following statement: the relation ρ is anti-symmetric.
(2 marks)
(e) Prove or disprove the following statement: the relation ρ is transitive.
(2 marks)
Page 7 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
7. (a) List the cyclic subgroups of (Z3 × Z3,+).
(5 marks)
(b) Explain why (Z3 × Z3,+) is not isomorphic to (Z9,+).
(2 marks)
Page 8 of 15
Semester One Final Examination, 2018 MATH1061 Discrete Mathematics
8. Solve the equation 12x + 15 = 11 in the field (Z53,+, ·). Hence determine the smallest positive
integer y such that 12y + 15 ≡ 11 (mod 53).
(Hint: 12× 31 = 7× 53 + 1.)
(3 marks)

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MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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