Math 212, Assignment 1

All questions are equally weighted. They will be marked for correctness and clarity of explanation.

1. For each of the following sets, determine if the given operation is a binary

operation or not. Explain your answers.

(a) The set of all 2×2 matrices with real entries whose second row is twice

its first row:

X =

a b

2a 2b

| a, b ∈ R

with matrix multiplication.

(b) Vectors in R

3 with dot product: (x, y, z) · (x

0

, y0

, z0

) = xx0 + yy0 + zz0

.

2. Let S be any set. Consider the binary operation intersection on the power

set of S, P(S). Is this operation associative? Is it commutative? Does it

have an identity? Find it, or explain why it doesn’t have one.

3. Prove that matrix multiplication is a binary operation on the set

X =

a b

c d

| a, b, c, d ∈ R, ad − bc 6= 0

.

(This set is GL2(R), but don’t use that here.)

4. Definition: Let X be a set with a binary operation, and let a be an element

of X. We say that an element b ∈ X is a left inverse of a if ba = e. We say

that b is a right inverse of a if ab = e. (Therefore b is an inverse of a if it

both a left inverse and a right inverse of a.)

Let A be a non-empty set and f : A → A. Prove that f has a right inverse

in FA if and only if f is surjective (onto).

5. For each of the following, determine (with proof) whether the given set is

a group with respect to the given operation.

(a) The set P of all polynomial functions from R to R with the operation

composition. That is, the set

P = {f : R → R | f(x) = a0 + a1x + · · · anx

n

, where a0, . . . , an ∈ R},

with composition.

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(b) The set R \ {−1} with the operation ∗ defined by

a ∗ b = a + b − ab.

(c) The power set P(S) of a set S, with the operation intersection.

6. Find, with proof, the order of

1 0

1 1

in GL2(R)?

7. Let k be a positive integer and let a be a negative integer. Prove using the

axioms for the integers that a

k = (−a)

k

if and only if k is even, using the

axioms for the integers. (You may also use the results from Chapter 1 and

the results from Chapter 2 that follow from the axioms.)

8. Let G be a finite group of even order. Prove that there is an element a 6= e

of G such that a

2 = e.

Rules for group assignments. Make sure you follow the universal rules for

group assignments (below) and any additional rules/procedures laid out in your

Group Contract.

1. Each group member is expected to contribute to the best of their ability, and

assignment submissions should only include the names of group members

who meet this expectation.

2. Each group member should be able to explain the group’s solution to me

and answer any questions I may have about it. It is the whole group’s

responsibility to ensure that this standard is met.

3. The task of composing final solutions and writing them up in good copy

must be shared equally among all group members (after a collaborative

problem-solving process).

4. After good copy solutions are complete, they should be shared among all

group members to be double-checked and proofread. This should be done

in advance of the due date, to allow time for any necessary corrections.

Corrections should be completed by the person who wrote the original solution.

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