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Math 120A Homework 1
Definitions Complete the following sentences.
1. A binary relation ∗ on a set X is associative if and only if
2. A binary relation ∗ on a set X is commutative if and only if
3. The table of a commutative binary relation has
(type of symmetry)
4. An element e ∈ X is an identity for the binary relation ∗ if
Questions Hand in questions 2, 4, 5, 6, 7(a,b) at discussion on Tuesday 6th October
1. Given the following binary relation table, calculate
∗
a
b
c
d
(i) c ∗ d
(ii) a ∗ (c ∗ b)
(iii) (c ∗ b) ∗ a
(iv) (d ∗ c) ∗ (b ∗ a)
a
c
d
a
b
b
d
c
b
a
c
a
b
c
d
d
b
a
d
c
2. Does it make sense to write a ∗ b ∗ c for the following binary relation ∗? Explain why/why not.
∗
a
b
c
a
b
c
b
b
c
a
a
c
b
a
c
3. Are the binary relations in 1,2 commutative? Explain your answer.
4.
(i) Exhibit a binary relation on the two element set { a, b} which is commutative.
(ii) Write down the multiplication tables of all non-commutative binary relations ∗ on { a, b}.
(iii) How many binary relations are there in total on { a, b}?
(iv) (Harder) How many distinct binary relations are there on a set of n letters? How many are
commutative?
5. For which of the following pairs is ∗ a binary relation on the set? For those that are relations,
which are commutative and which are associative?
(i) (Z, ∗), a ∗ b = a − b.
(ii) (Q+ , ∗), a ∗ b = ab .
(iii) (Z+ , ∗), a ∗ b = ab .
(iv) (Z+ , ∗), a ∗ b = product of all prime factors of ab (i.e. for 360 = 23 · 32 · 5 and 10 = 2 · 5 we
have 360 ∗ 10 = 2 · 3 · 5).
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6. A monoid is a set with an associative binary relation and an identity element. Which of the
structures in question 5 are monoids?
7. This question deals with sets of functions. The notation C means continuous and C1 means
differentiable functions with continuous first derivative. The domain is also indicated. Thus
C1 [0, 1] is the set of differentiable functions with continuous first derivatives, and domain [0, 1].
(a) Define ∗ on C1 [0, 1] by
( f ∗ g)( x ) =
Z x
0
f 0 (t) g0 (t) dt + f (0) + g(0)
Is ∗ a binary relation? Is it commutative? Associative? Prove your assertions.
(You will need to use the Fundamental Theorem of Calculus. . . )
(b) Suppose that e ∈ C1 [0, 1] is an identity for ∗. Show that e would have to satisfy the integral
equation
Z x
0
f 0 (t)e0 (t) dt + f (0) + e(0) = f ( x )
for all functions f ∈ C1 [0, 1]. Does such an e exist? Is it unique?
(c) (Harder) Define ∗ by
(
f if max f ≥ max g,
f ∗g=
g if max g > max f .
Explain why ∗ is a binary relation on C [0, 1], but isn’t on C (0, 1). Is (C [0, 1], ∗) a monoid?
(You’ll need to recall some elementary analysis: no excuse if you’ve taken Math 140A!)
8. The Lie bracket [ , ] of two n × n matrices is defined by [ A, B] = AB − BA.
(i) Show that [ , ] is an anti-commutative binary relation on Mn (R), that is,
∀ A, B ∈ Mn (R),
[ A, B] = −[ B, A]
Give a counter-example with 2 × 2 matrices which shows that the Lie bracket is nonassociative.
(ii) Let so(n) = { A ∈ Mn (R) : A T = − A} be the set of skew-symmetric n × n matrices.
i. Is matrix multiplication a binary relation on so(n)?
ii. Is the Lie bracket a binary relation on so(n)?
Justify your answers. If you like, try proving this first with 2 × 2 matrices, then see if you can
generalize your arguments. You will only need the abstract definitions, not any explicit matrices.
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