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ALGEBRA AND LOGIC
创建日期：2024-05-21 14:49:52
所属分类：数学mathematics
标签： MATH3066
MATH3066 ALGEBRA AND LOGIC

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MATH3066 ALGEBRA AND LOGIC

Semester 1 First Assignment 2023
This assignment comprises 60 marks and is worth 20% of the overall assessment.
It should be completed and uploaded into Canvas before midnight on Friday 31
March 2023. Acknowledge any sources or assistance. This must be your own work.
Breaches of academic integrity, including copying solutions, sharing answers and
attempts at contract cheating, attract severe penalties.
1. Construct a truth table for the following well-formed formula W in the propo-
sitional variables P , Q and R and find all counterexamples.
W ≡ (P ∨ Q) ⇒ ∼ (Q ∧ R )
(5 marks)
2. Find a well-formed formula W in the propositional variables P , Q and R, using
at most two instances of binary connectives, having the following truth table:
P Q R | W
|
T T T | F
T T F | F
T F T | F
T F F | T
F T T | T
F T F | T
F F T | T
F F F | T
Justify your answer. Prove that it is impossible to represent W using at most
one instance of a binary connective.
(10 marks)
3. Use the rules of deduction in the Propositional Calculus to find formal proofs
for the following sequents. You have access to the 10 basic rules, and use of
definition of the double arrow for the last sequent, but should not use sequent
or theorem introduction.
(a) (P ∨ Q) ⇒ R ` (P ⇒ R) ∧ (Q ⇒ R)
(b) P ⇒ Q , ∼ (∼ R ∧ S) ` (Q⇒ S) ⇒ (P ⇒ R)
(c) P ⇔ Q ` (P ∨ Q) ⇔ (P ∧ Q)
(15 marks)
4. Use truth values to determine which one of the following is a theorem (in the
sense of always being true).
(a)
(
P ∧ (Q ∨ ∼ P )) ⇒ Q (b) (Q ∧ (Q ∨ ∼ P )) ⇒ P
For the one that isn’t a theorem, produce a counterexample. For the one
that is a theorem, provide a formal proof also using rules of deduction in the
Propositional Calculus (but avoiding derived rules).
(10 marks)
5. Consider well-formed formulae Wn, for each positive integer n, defined as fol-
lows, where P1, P2, P3, . . . are propositional variables:
W1 = ∼ P1 and Wk+1 = ∼ (∼Wk ∨ Pk+1) for each k ≥ 1 .
Prove, for each positive integer n, that Wn has truth value T if and only if
P1, . . . , Pn all have truth value F .
(5 marks)
6. Prove that every well-formed formula in the Propositional Calculus is equivalent
to a well-formed formula where the only connectives used are ∼ and ⇒ . You
may use without proof the fact that if two well-formed formulae are logically
equivalent, then replacing one by the other as subwffs in any given well-formed
formula retains logical equivalence.
(8 marks)
7. Consider the following equations over Z:
112 = 2(39) + 34 ,
39 = 34 + 5 ,
34 = 6(5) + 4 ,
5 = 4 + 1 .
(a) Use these equations to find integers α and β such that
112α+ 39β = 1 .
(b) Use the previous part to find the multiplicative inverse of 39 in Z112 and
the multiplicative inverse of 34 in Z39.
(7 marks)
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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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