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ms Sets and Logic

Coursework 2B (2012–201

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Notes

Notes

1. The module code is 124ms, and the module title is Sets and Logic.

2. our assignment should be submitted at the hand-in point in the Engineering an

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Computing Building by 16:00 on 15th April 2013. Note that work handed in later than

this without attached evidence of an extension/deferral will be given a mark of 0.

3. Work should be secured by a staple at the top left hand corner. Do not put it in a plastic

wallet or a binder. If it is printed, it should be double-sided.

4. Remember that you must have a barcoded cover sheet and two copies of your

assignment, one of which you submit along with the cover sheet, and the other for

your own records.

5. This is an individual assignment, not a group assignment. Collaborating on the

assignment will be regarded as cheating.

6. This assignment is worth 60% of the module mark.

2 Assessment Criteri

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To obtain a first class mark, you should submit work that is technically correct (with at most

minor slips) and is fully explained in grammatically correct standard English. (Due

consideration will be made for students with relevant registered disabilities, such as

dyslexia.)

So for example, notation should be correct, and any deviations from the notation used in

class should be explained. If you present a truth or membership table, part of the credit will

be for explaining where the table comes from, part for correct entries, and part for an

explanation of how the table does its job. If you present a formal proof, you should clearly

explain what are the hypotheses, and what is the conclusion; the proof should be presented

in the standard (tabular) manner, and you should give a brief explanation to show that the

proof does what it is supposed to. If you use a standard algorithm, you should explain what

happens at each step.

A bare pass would normally be awarded for a submission which shows that a serious

attempt, with at least some success, has been made to understand the relevant mathematics

and apply it appropriately.

A marks breakdown is shown on the assignment, and a full worked solution and marking

scheme will be provided after the hand-in deadline.

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Attempt ALL seven questions

1. The functions f : {a, b, c}→{1, 2, 3} and g : {1, 2, 3}→{x, y} are given by f (a) = 3,

f (b) = 1, f (c) = 2, g(1) = x, g(2) = x, g(3) = y.

(a) Classify each of f and g as bijective, injective, surjective, or neither.

(b) Find g ◦ f .

(c) Either find the inverse of g ◦ f or explain why g ◦ f is not invertible.

(8 marks)

2. I want to develop a database of information about my book collection, which tells me

about the authors and genres of the various books I own. At the moment I have books

by Isaac Asimov, China Mieville, Peter F Hamilton and Arthur C Clarke, and I denote

the set of authors by A = {A, M, H, C}, abbreviating each author by the initial of his

surname. I am classifying the books as science fiction, fantasy, horror and non-fiction,

so my set of genres is G = {s, f , h, n}, again using initial letters as abbreviations.

At the moment, I have science fiction works by Isaac Asimov, Arthur C Clarke, and

Peter F Hamilton, I have fantasy by Peter F Hamilton and Isaac Asimov, horror by

Peter F Hamilton, and non-fiction by Isaac Asimov and Arthur C Clarke.

(a) Give the relation R on A×G which represents this information. (You may use

appropriate abbreviations.)

(b) Find the combination of projection and inverse projection maps which finds all

authors by whom I have non-fiction books.

(c) Find the combination of projection and inverse projection maps which find all

writers who have written fantasy but not horror books in my collection.

(8 marks)

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3. (a) Draw the graph with adjacency matrix

A =

0 1 0 0 1

1 0 1 0 0

0 1 0 1 1

0 0 1 0 1

1 0 1 1 0

where the columns and rows label vertices 1 to 5 in order.

(b) Use the adjacency matrix connectivity algorithm, starting by marking row 2 and

crossing out column 2, to show whether this graph is connected.

(c) Calculate A2 and hence find the number of paths of length 2 from vertex 2 to

vertex 4.

(d) Use breadth first search starting at vertex 1 to find a spanning tree.

(11 marks)

4. Use Dijkstra’s algorithm to find the shortest path from node a to node d in the

following graph.

a

b c

d

ef

2

2

2

6

7

3

5

1

(8 marks)

5. Use the heapsort algorithm to put the following list of numbers in increasing order:

8 2 1 4 3 7 5

You should explain in detail how the original heap is obtained, and then show your

sequence of heaps and partial ordered lists. (8 marks)

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6. Consider the symbols and frequencies:

o : 4 e : 14 n : 6 t : 10 s : 8 m : 3

(a) Find a Huffman code for this situation, and the average length of an encoded

symbol.

(b) Assign the symbols to these codewords in a different order, and comment on the

resulting average length.

(9 marks)

7. Bob decides to use n = 187 = 11 × 17 and e = 23 as his public key for an RSA

cryptosystem.

(a) Show that the decryption exponent is 7.

(b) Find the encrypted form of the message 37.

(8 marks)

RJL,124ms\coursework\cw2b.tex

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