complete ALL seven questions.

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 Please see attachment to complete ALL seven questions.

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20140707074554cw2jan.pdf

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124ms Sets and Logic Coursework 2 (2013–2014) 1 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 and Computing Building by 16:00 on 23th July 2014. 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 Criteria 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. 1 Attempt ALL seven questions 1. The functions f : {1, 2, 3} → { a, b} and g : { a, b} → { x, y, z} are given by f (1) = b, f (2) = b, f (3) = a, g( a) = y, g(b) = x. (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 China Mieville and Isaac Asimov, I have fantasy by Peter F Hamilton and China Mieville, horror by Peter F Hamilton and Arthur C Clarke, and non-fiction by Isaac Asimov. (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 horror books. (c) Find the combination of projection and inverse projection maps which find all writers who have written non-fiction or fantasy books in my collection. (8 marks) 2 3. (a) Draw the graph with adjacency matrix    A=   0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 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 1 and crossing out column 1, to show whether this graph is connected. (c) Calculate A2 and hence find the number of paths of length 2 from vertex 1 to vertex 5. (d) Find a breadth first spanning tree starting at vertex 3. (11 marks) 4. Use Dijkstra’s algorithm to find the shortest path from node a to node f in the following graph. c 3 b 3 5 2 7 a d 12 2 4 f e (8 marks) 5. Use the heapsort algorithm to put the following list of numbers in decreasing order: 8 2 4 7 1 3 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) 3 6. Consider the symbols and frequencies: o : 12 e : 10 n : 4 t : 6 s : 5 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 = 221 = 13 × 17 and e = 11 as his public key for an RSA cryptosystem. (a) Show that the decryption exponent is 35. (b) Find the encrypted form of the message 16. (8 marks) RJL,124ms\coursework\cw2jan.tex 4 124ms Sets and Logic Coursework 2 (2013–2014) 1 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 and Computing Building by 16:00 on 23th July 2014. 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 Criteria 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. 1 Attempt ALL seven questions 1. The functions f : {1, 2, 3} → { a, b} and g : { a, b} → { x, y, z} are given by f (1) = b, f (2) = b, f (3) = a, g( a) = y, g(b) = x. (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 China Mieville and Isaac Asimov, I have fantasy by Peter F Hamilton and China Mieville, horror by Peter F Hamilton and Arthur C Clarke, and non-fiction by Isaac Asimov. (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 horror books. (c) Find the combination of projection and inverse projection maps which find all writers who have written non-fiction or fantasy books in my collection. (8 marks) 2 3. (a) Draw the graph with adjacency matrix    A=   0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 1 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 1 and crossing out column 1, to show whether this graph is connected. (c) Calculate A2 and hence find the number of paths of length 2 from vertex 1 to vertex 5. (d) Find a breadth first spanning tree starting at vertex 3. (11 marks) 4. Use Dijkstra’s algorithm to find the shortest path from node a to node f in the following graph. c 3 b 3 5 2 7 a d 12 2 4 f e (8 marks) 5. Use the heapsort algorithm to put the following list of numbers in decreasing order: 8 2 4 7 1 3 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) 3 6. Consider the symbols and frequencies: o : 12 e : 10 n : 4 t : 6 s : 5 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 = 221 = 13 × 17 and e = 11 as his public key for an RSA cryptosystem. (a) Show that the decryption exponent is 35. (b) Find the encrypted form of the message 16. (8 marks) RJL,124ms\coursework\cw2jan.tex 4
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