Introduction to Abstract Algebra/Problem set 2

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 $$f(x) = x^2$$.
 * $$f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$$ such that $$f(x) = x^2$$.
 * $$f:\mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(x) = x^2$$.
 * $$f:\mathbb{Z} \rightarrow \mathbb{Z}$$ such that $$f(x) = 2x$$.

2: Prove that if $$A$$ and $$B$$ are nonempty sets, then the function $$f: A \times B \rightarrow B \times A$$ given by the relation $$f((a, b)) = (b, a)$$ is a bijection.

3: Suppose the set $$A$$ is finite.
 * Prove that if $$f: A \rightarrow A$$ is an onto map, then $$f$$ is a one-to-one map.
 * Prove that if $$f: A \rightarrow A$$ is a one-to-one map, then $$f$$ is an onto map.

4: Suppose that the set $$A$$ is not finite.
 * Provide a counter example to the proposition that if $$f: A \rightarrow A$$ is an onto map, then $$f$$ is a one-to-one map.
 * Provide a counter example to the proposition that if $$f: A \rightarrow A$$ is a one-to-one map, then $$f$$ is an onto map.