Homework 9

1. Prove or disprove that the function $x \mapsto 2x$ from $\mathbb{R}$ to $\mathbb{R}$ is surjective.

2. Prove or disprove that the function $x \mapsto 2x$ from $\mathbb{Z}$ to $\mathbb{Z}$ is surjective.

3. Prove or disprove that for all functions $f:\mathbb{Q}\to\mathbb{Q}$, if $f$ is injective then the function $x\mapsto f(x) + 1$ is injective.

4. Prove or disprove that the function $x \mapsto 3-2x$ from $\mathbb{R}$ to $\mathbb{R}$ is surjective.

5. Consider the function $(x,y,z)\mapsto (x+y+z,x+2y+3z)$ from $\mathbb{R}^3$ to $\mathbb{R}^2$. Prove or disprove that this function is injective.

6. Consider the function $(r,s)\mapsto (s, r+2s)$ from $\mathbb{Q}^2$ to $\mathbb{Q}^2$. Prove or disprove that this function is bijective.

All problems on this homework are "prove or disprove" problems, which means that you need to discover for yourself whether the given statement is true or false, then only prove it (if true) or only disprove it (if false).