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MATH7861 ASSIGNMENT
创建日期：2024-06-04 18:07:58
所属分类：数学mathematics
标签： MATH7861
ASSIGNMENT

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MATH7861 ASSIGNMENT 

Marking. Each question is worth 10 marks. Total 80 marks.
Submission. Please submit online via blackboard. You must submit ONE PDF file with your
NAME, STUDENT NUMBER and TUTORIAL GROUP NUMBER clearly at the top of
the first page. Make sure that your pdf file is legible and that the file size is not excessive. If you take
photos with your phone, the file size will be very large, and you must compress the size.1
Help. You can get help for doing the questions but solutions must be written in your own words.
Copying from any source (including AI) constitutes cheating and will be detrimental to your studies
at UQ.
Problems.
(1) (10 marks) Define f : {1, 2} × Z→ Z by f(a, b) := 2b+ a.
Determine the following (make sure to justify your answers):
(a) (2 marks) Domain of f .
(b) (2 marks) Range of f .
(c) (2 marks) Is f one to one?
(d) (2 marks) Is f onto?
(e) (2 marks) Is f bijective?
(2) (10 marks) Prove that for every positive integer n, and sets A and B1, B2, . . . , Bn
A−
n⋂
i=1
Bi =
n⋃
i=1
(A−Bi).
(3) (10 marks) Let X be a set and (Y,≤) a partially ordered set. Let F denote the set of functions
from X to Y . Define a relation ≤ on F by setting f ≤ g if f(x) ≤ g(x) for all x ∈ X. Show
that ≤ is a partial ordering on F .
(4) (10 marks) Suppose X, Y are two disjoint countable sets. Prove that X ∪ Y is also countable.
1See, for instance, https://pdfcompressor.com for compressing tools.
1
(5) (10 marks) Let f be a function from the set A to the set B. Let S and T be subsets of A.
(a) (2 marks) Prove that f(S ∪ T ) = f(S) ∪ f(T ).
(b) (2 marks) Prove that f(S ∩ T ) ⊆ f(S) ∩ f(T ).
(c) (3 marks) Give an example of sets A, B, S, T and a function f for which f(S ∩ T ) ⊂
f(S) ∩ f(T ).
(d) (3 marks) Prove that if f is one-one, then f(S ∩ T ) = f(S) ∩ f(T ).
(6) (10 marks) Let S be the set of all binary strings of length 6. Consider the relation ρ on the
set S in which for all a, b ∈ S, (a, b) ∈ ρ if and only if the length of a longest substring of
consecutive ones in a is the same as the length of a longest substring of consecutive ones in b.
(a) (2 marks) Is 011010 related to 000011? Explain why or why not.
(b) (6 marks) Prove that ρ is an equivalence relation.
(c) (2 marks) List the elements of the equivalence class [111100].
(7) (10 marks) Find an explicit bijection (i.e. give a formula that explains what f(x) is given any
x ∈ [0, 1]) between [0, 1] and [0, 1).
(8) (10 marks) Define a relation R on Z by xRy if x2 − y2 is divisible by 4. Is R an equivalence
relation? Justify your answer. If it is an equivalence relation, determine the equivalence classes.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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