Math 220 Assignment 7
Due November 6th
Recall: given a set A with a binary operator ∗, and a subset S ⊆ A, we say
S is closed under ∗ if, ∀σ, τ ∈ S, we have σ ∗ τ ∈ S.
1. Let f : A → B, and let S ⊆ A. Define f |S : S → B by f |S (σ) = f (σ), for
all σ ∈ S.
(a) Prove that if f is injective, then f |S is injective.
(b) Disprove the converse of (a). This means finding an example of a
function f : A → B and a set S ⊆ A such that f is not injective, but
f |S is injective.
(c) Show that if f |S is surjective, then so is f .
(d) Disprove the converse of (b). This means finding an example of
f : A → B, and S ⊆ A such that f is surjective, but f |S is not.
2. Let A be a non-empty set, and let f : A → A, g : A → A, and h : A → A
be functions. Suppose f ◦ g and g ◦ h are bijections. Prove that f, g and
h are all bijections. Hint: there are multiple approaches, but using the
previous homework assignment certainly helps.
3. Let ∗ be a associative binary operation on A, and suppose there is an
identity element e for ∗. Let a, b ∈ A, and suppose a−1 , b−1 exists. Prove
that (ab)−1 = b−1 a−1 . Hint: the definition of inverse suggests that you
need to show that (b−1 a−1 )(ab) = e = (ab)(b−1 a−1 ).
√
√
4. Let Q[ 3] = {a + b 3 : a, b ∈ Q}.
√
(a) Show that Q[
√ 3] is closed
√ under multiplication. In other words, for
all x, y ∈ Q[ 3], xy ∈ Q[ 3].
√
√
3
(b) Given a+b 3, show that its multiplicative inverse is given by aa−b
2 −3b2 .
Nice fact: these results
√ remain true if you replace 3 with any rational
number D such that D is irrational.
5. Let A be a set. Then interesection ∩ forms a binary operator on P (A).
That is, for all X, Y ⊆ A, X ∩ Y ⊆ A. Does this operator have an identity
element? Which subsets of P (A) have inverses with respect to ∩? Prove
your answers.
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