1.

(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. Consider the group of symmetries of a square. The symmetries of a square are:

i-: the identity,

r₁: clockwise rotation through 90⁰,

r₂: clockwise rotation through 180⁰,

r₃: clockwise rotation through 270⁰,

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.

3. 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. 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. 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?