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Math 212, Assignment 1

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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.
1
(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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PlaceholderMath 212, Assignment 1
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