Recent Question/Assignment
Semester 2, 2015 MATH1061 Assignment 5
Assignment 5 must be submitted by 10 am on Monday 26th of October, 2015. For MATH1061 students, this assignment is worth 4% of your final mark. Assignments must be submitted through the electronic assignment submission system on level 3 of building 67 (unless you are enrolled as an external student). To use the system you must print the coversheet that will be emailed to you approximately 7 days before the assignment is due, and staple that coversheet to your assignment, ensuring that your tutorial group appears on the coversheet.
1. (7 marks)
(a) Prove that the set of bijections from Z to Z with the operation composition of functions is a group. You may assume that composition of functions is associative.
(b) Is this group abelian?
2. (7 marks) Consider the group of symmetries of a square. The symmetries of a square are:
?: the identity, r1: clockwise rotation through 90?, r2: clockwise rotation through 180?, r3: clockwise rotation through 270?, s1: reflection in the line passing through the midpoints of the top and bottom edges, s2: reflection in the line passing through the midpoints of the left and right edges, s3: reflection in the line passing through the top left and bottom right corners, s4: reflection in the line passing through the top right and bottom left corners.
(a) Write out the Cayley table for this group, with the headline and sideline in the order
?,r1,r2,r3,s1,s2,s3,s4.
(b) List all the subgroups of order 2 in the group of symmetries of a square.
(c) Determine o(r3) in the group of symmetries of a square.
(d) Is the group of symmetries of a square a cyclic group? Explain your answer.
Continues next page
Semester 2, 2015 MATH1061 Assignment 5
3. (6 marks) Let A be the set of all 5 digit numbers whose digits are all odd, that is, 5 digit numbers created using the digits 1,3,5,7,9 (repetition allowed). Let B be the set of all 5 digit numbers whose digits are all odd and all distinct, that, is 5 digit numbers created using the digits 1,3,5,7,9 in which no digit is used more than once.
(a) Determine |A|.
(b) Determine |B|.
(c) If I chose an element of B at random, what is the probability that the number I’ve chosen is divisible by 5 ?
4. (4 marks) In a certain discrete math class, three quizzes (each marked out of 15) were given. Out of the 35 students in the class: 15 students scored 12 or above on quiz #1,
12 students scored 12 or above on quiz #2,
18 students scored 12 or above on quiz #3,
7 students scored 12 or above on quizzes #1 and #2,
11 students scored 12 or above on quizzes #1 and #3,
8 students scored 12 or above on quizzes #2 and #3,
4 students scored 12 or above on quizzes #1, #2 and #3.
(a) How many students scored 12 or above on at least one quiz?
(b) How many students scored 12 or above on quizzes 1 and 2 but not on quiz 3 ?
5. (8 marks) A quiz consists of ten True/False questions. Each question is answered with either True or False (no question is left blank).
(a) How many possible sequences of answers are there to the ten questions?
(b) How many of the possible answer sequences begin and end with the answer True?
(c) How many of the possible answer sequences have the correct answer to precisely 5 questions?
(d) If a student guesses at all ten questions, what is the probability that they guess the correct answer to at least 8 out of the 10 questions?
Assignment 5 must be submitted by 10 am on Monday 26th of October, 2015. For MATH1061 students, this assignment is worth 4% of your final mark. Assignments must be submitted through the electronic assignment submission system on level 3 of building 67 (unless you are enrolled as an external student). To use the system you must print the coversheet that will be emailed to you approximately 7 days before the assignment is due, and staple that coversheet to your assignment, ensuring that your tutorial group appears on the coversheet.
1. (7 marks)
(a) Prove that the set of bijections from Z to Z with the operation composition of functions is a group. You may assume that composition of functions is associative.
(b) Is this group abelian?
2. (7 marks) Consider the group of symmetries of a square. The symmetries of a square are:
?: the identity, r1: clockwise rotation through 90?, r2: clockwise rotation through 180?, r3: clockwise rotation through 270?, s1: reflection in the line passing through the midpoints of the top and bottom edges, s2: reflection in the line passing through the midpoints of the left and right edges, s3: reflection in the line passing through the top left and bottom right corners, s4: reflection in the line passing through the top right and bottom left corners.
(a) Write out the Cayley table for this group, with the headline and sideline in the order
?,r1,r2,r3,s1,s2,s3,s4.
(b) List all the subgroups of order 2 in the group of symmetries of a square.
(c) Determine o(r3) in the group of symmetries of a square.
(d) Is the group of symmetries of a square a cyclic group? Explain your answer.
Continues next page
Semester 2, 2015 MATH1061 Assignment 5
3. (6 marks) Let A be the set of all 5 digit numbers whose digits are all odd, that is, 5 digit numbers created using the digits 1,3,5,7,9 (repetition allowed). Let B be the set of all 5 digit numbers whose digits are all odd and all distinct, that, is 5 digit numbers created using the digits 1,3,5,7,9 in which no digit is used more than once.
(a) Determine |A|.
(b) Determine |B|.
(c) If I chose an element of B at random, what is the probability that the number I’ve chosen is divisible by 5 ?
4. (4 marks) In a certain discrete math class, three quizzes (each marked out of 15) were given. Out of the 35 students in the class: 15 students scored 12 or above on quiz #1,
12 students scored 12 or above on quiz #2,
18 students scored 12 or above on quiz #3,
7 students scored 12 or above on quizzes #1 and #2,
11 students scored 12 or above on quizzes #1 and #3,
8 students scored 12 or above on quizzes #2 and #3,
4 students scored 12 or above on quizzes #1, #2 and #3.
(a) How many students scored 12 or above on at least one quiz?
(b) How many students scored 12 or above on quizzes 1 and 2 but not on quiz 3 ?
5. (8 marks) A quiz consists of ten True/False questions. Each question is answered with either True or False (no question is left blank).
(a) How many possible sequences of answers are there to the ten questions?
(b) How many of the possible answer sequences begin and end with the answer True?
(c) How many of the possible answer sequences have the correct answer to precisely 5 questions?
(d) If a student guesses at all ten questions, what is the probability that they guess the correct answer to at least 8 out of the 10 questions?
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