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students this assignment is worth 10% of your final mark.
1. (8 marks) We let M2×2(R) denote the set of 2× 2 matrices
A =
(
a11 a12
a21 a22
)
with a11, a12, a21, a22 ∈ R. This set is given a binary operation · known as matrix multiplication
defined by (
a11 a12
a21 a22
)
·
(
b11 b12
b21 b22
)
=
(
a11b11 + a12b21 a11b12 + a12b22
a21b11 + a22b21 a21b12 + a22b22
)
.
Let O(2,R) denote the subset of M2×2(R) consisting of those matrices
A =
(
a11 a12
a21 a22
)
so that (
a11 a12
a21 a22
)
·
(
a11 a21
a12 a22
)
=
(
1 0
0 1
)
=
(
a11 a21
a12 a22
)
·
(
a11 a12
a21 a22
)
.
(a) Show that (O(2,R), ·) is a group. You may take as given that O(2,R) is closed under the
matrix multiplication operation ‘·’.
(b) Show that ({(−1 0
0 −1
)
,
(
0 −1
1 0
)
,
(
1 0
0 1
)
,
(
0 1
−1 0
)}
, ·
)
is a subgroup of (O(2,R), ·) which is isomorphic to (Z4,+).
2. (6 marks) Let (G1,+) and (G2,+) be two subgroups of (R,+) so that Z+ ⊆ G1 ∩ G2. If
φ : G1 → G2 is a group isomorphism with φ(1) = 1, show that φ(n) = n for all n ∈ Z+. Hint:
consider using mathematical induction.
3. (16 marks) Given a prime number k, we define Q(
√
k) = {a + b√k : a, b ∈ Q} ⊆ R. This set
becomes a field when equipped with the usual addition and multiplication operations inherited
from R.
(a) For each non-zero x ∈ Q(√2) of the form x = a+ b√2, prove that x−1 = a
a2−2b2 − ba2−2b2
√
2.
(b) Show that
√
2 /∈ Q(√3). You can use, without proof, the fact that √2,√3,
√
2√
3
are all
irrational numbers.
(c) Show that there cannot be a function φ : Q(
√
2)→ Q(√3) so that
φ : (Q(
√
2)− {0},×)→ (Q(
√
3)− {0},×)
and
φ : (Q(
√
2),+)→ (Q(
√
3),+)
are both group isomorphisms. Hint: What can you say about φ(
√
2×√2)?
4. (9 marks) Suppose that we have 100 apples. In order to determine the integrity of the entire
batch of apples, we carefully examine n randomly-chosen apples; if any of the apples is rotten,
the whole batch of apples is discarded. Suppose that 50 of the apples are rotten, but we do not
know this during the inspection process.
(a) Calculate the probability that the whole batch is discarded for n = 1, 2, 3, 4, 5, 6.
(b) Find all values of n for which the probability of discarding the whole batch of apples is at
least 99% = 99
100
.
5. (15 marks) Short essay on
– the Gambler’s ruin, or
– the Monty Hall problem.
You are asked to choose from the above topics and write a 1.5 to 2.5 page essay that discusses the
related theory and applications of the topic you have chosen. Your discussion should include his-
torical aspects of the topic and how this topic relates to the mathematics studied in MATH7861.
Examples and diagrams of the relevant mathematics should be included. Some references to get
you started on the topic are available through the blackboard site.
Essay Structure
The essay should be TYPED with a minimum of 1.5 pages and a maximum of 2.5 pages including
a bibliography. The font should be 12pt and margins should be around 1.5 cm. Overall the length
of your essay should be around 1500 words, with diagrams to support your text. You can (but
not required to) follow the following structure for you essay:
Background+History (1-2 paragraphs):
In this section you can introduce the main subject of this essay, discussing the historical aspects
of the subject. (4 marks)
Analysis Section (3-4 paragraphs): What are the mathematical ideas associated with this
topic and how do they relate to the mathematics being studied in MATH7861? Can you give
examples of how this mathematics is relevant in our ever day lives? (5 marks)
Conclusion (1 paragraph): Conclude the essay by summarizing the ideas presented, if possible
discuss the impact of this mathematics on your current studies. (3 marks)
Bibliography: At least 3 sources (you can include the supplied article as a reference) correctly
cited in the essay and using a consistent citation format. (3 marks)