advertisement

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. 1