Homework 9

1. Let $f : X \to Y$ and $g : Y \to Z$ be injective functions. Show that $g \circ f$ is also injective.

2. Prove or disprove that the function $x \mapsto 5+3x$ from $\mathbb{R}$ to $\mathbb{R}$ is bijective.

   (If you think it's true, prove it, by solving the first version in the Lean template. If you think it's false, solve the second version.)

3. Consider the function $(x,y,z)\mapsto (x+y+z,x-2y+z)$ from $\mathbb{R}^3$ to $\mathbb{R}^2$. Show that this function is not injective.

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

   Do this following the structure in the Lean template, by providing an inverse for the function.

5. Consider the function $(x,y)\mapsto x^2+y^2$ from $\mathbb{Q}^2$ to $\mathbb{Q}$. Show that this function is not surjective.

6. Consider the function $(x,y)\mapsto x^2-y^2$ from $\mathbb{Q}^2$ to $\mathbb{Q}$. Show that this function is surjective.