Introduction to Abstract Algebra/Problem set 2

From Wikiversity

Jump to: navigation, search
Information icon.svg This page is uncategorized.

Please help improve this article by adding it to one or more categories, so it may be associated with related articles (how?). Please remove this tag after categorizing, but not before. When categorizing this resource, the green project boxes on the right will guide you to suitable help pages for categorizing by subject, educational level, resource type and completion status. You can also add further categories of your choice.

Problem Set #2: Introduction to Abstract Algebra.

As you work through these problems, think about the logical steps you are using. You should know if your proof is correct or not if you have a reason for every step.

1: Determine if the follow maps are onto and/or 1:1:

  • f:\mathbb{R} \rightarrow \mathbb{R}^+ such that f(x) = x2
  • f:\mathbb{R}^+ \rightarrow \mathbb{R}^+ such that f(x) = x2
  • f:\mathbb{Z} \rightarrow \mathbb{Z} such that f(x) = x2
  • f:\mathbb{Z} \rightarrow \mathbb{Z} such that f(x) = 2x

2: Prove that if A and B are nonempty sets, then f: A x B \rightarrow B x A is a bijection.

3: Suppose the set A is finite.

  • Prove if f: A \rightarrow A, then f is a 1:1 map.
  • Prove if f is a 1:1 map from A to A, then f is an onto map.
Retrieved from "http://en.wikiversity.org/wiki/Introduction_to_Abstract_Algebra/Problem_set_2"