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COMP 9020 – Boolean Algebra Assignment

创建日期：2024-06-25 17:41:07

所属分类：数学mathematics

标签： COMP 9020

Boolean Algebra

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COMP 9020 – Assignment 3

Note: In your assignment, how you arrived at your solution is as important (if not more so) than the solution itself and will be assessed accordingly. There may be more than one way to find a solution, and your approach should contain enough detail to justify its correctness. Lecture content can be assumed to be common knowledge.

1. Let (T, ∧, ∨,j , 0, 1) be a Boolean Algebra.

Define ∗ : T × T → T and ◦ : T × T → T as follows:

x ∗ y := (x ∨ y)j x ◦ y := (x ∧ y)j

(a) Show, using the laws of Boolean Algebra, how to define x ∗ y using only x, y, ◦ and parentheses.

(b) Show, using the laws of Boolean Algebra, how to define x ◦ y using only x, y, ∗ and parentheses.

(10 marks)

Define R ⊆ T × T as follows:

(x, y) ∈ R if, and only if, (x ∧ y) ∨ (xj ∧ yj) = 1

(c) Show, using the laws of Boolean Algebra, that R is an equivalence relation. Hint: You may want to use the observation that if A = B = 1 then A ∧ B ∧ C = A ∧ B implies C = 1 (why?)

(10 marks)

2. Let P F denote the set of well-formed propositional formulas made up of propositional variables, , , and the connectives , , and . Recall from Quiz 7 the definitions of dual and flip as functions from P F to P F :

• dual(p) = p • flip(p) = ¬p

• dual(T) = ⊥; dual(⊥) = T • flip(T) = T; flip(⊥) = ⊥

• dual(¬ϕ) = ¬dual(ϕ) • flip(¬ϕ) = ¬flip(ϕ)

• dual(ϕ ∧ ψ) = dual(ϕ) ∨ dual(ψ) • flip(ϕ ∧ ψ) = flip(ϕ) ∧ flip(ψ)

• dual(ϕ ∨ ψ) = dual(ϕ) ∧ dual(ψ) • flip(ϕ ∨ ψ) = flip(ϕ) ∨ flip(ψ)

(a) For the formula ϕ = p ∨ (q ∧ ¬r):

(i) What is dual(ϕ)?

(ii) What is flip(ϕ)?

(b) Prove that for all ϕ ∈ P F : flip(ϕ) is logically equivalent to ¬dual(ϕ).

(10 marks)

3. Let P (n) be the proposition that: for all k, with 1 ≤ k ≤ n,

.nΣ = .n − 1Σ + .n − 1Σ.

k k − 1 k

(a) Prove that P (n) holds for all n 1. (Note: it is possible to do this without using induction)

(10 marks)

We can compute n from the formula given in lectures, however this can often require computing unnecessarily large numbers. For example,

100 = 253338471349988640 which can be expressed as a 64-bit integer,

but 100! is larger than a 512-bit integer. We can, however, make use of the equation above to compute n more efficiently. Here are two algorithms

for doing this:

chooseRec(n, k) : chooseIter(n, k) :

if k = 0 or k = n : return 1 Let C be a n n array else : for m = 1 to n :

x := chooseRec(n 1, k 1) C[m][0]=C[m][m]=1

y := chooseRec(n 1, k) for j = 1 to m 1 :

return x + y C[m][j]=C[m 1][j 1]

+ C[m 1][j]

return C[n][k]

Let trec(n, k) be the running time for chooseRec(n, k), and let titer(n) be the running time for chooseIter(n, k). Let Trec(n) = max0≤k≤n trec(n, k) and Titer(n) = max0≤k≤n titer(n, k) (so Trec(n) ≥ trec(n, k) for all k, and likewise for Titer(n)).

(b) Give an asymptotic upper bound for Trec(n). Justify your answer.

(10 marks)

Advice on how to do the assignment

All submitted work must be done individually without consulting someone else’s solutions in accordance with the University’s “Academic Dishonesty and Pla- giarism” policies.

Assignments are to be submitted via WebCMS (or give) as a single pdf (max size 2Mb). In Linux, the following command

pdfjoin –outfile output.pdf input1.pdf input2.pdf …

can be used to combine multiple pdf files. The command

convert -density 150×150 -compress jpeg input.pdf output.pdf

can be used to reduce the filesize of a pdf (change 150 to reduce/improve quality/filesize). Please ensure your files are legible before submitting.

Be careful with giving multiple or alternative answers. If you give multiple answers, then we will give you marks only for ”your worst answer”, as this indicates how well you understood the question.

Some of the questions are very easy (with the help of the lecture notes or book). You can use the material presented in the lecture or book (without proving it). You do not need to write more than necessary (see comment above).

When giving answers to questions, we always would like you to prove/exp- lain/motivate your answers.

If you use further resources (books, scientific papers, the internet,…) to formulate your answers, then add references to your sources.

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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 MGMTMFE 405 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