Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

1.

2.

Now consider the statement:

For all $y\in B\text{,}$ there exists an $x\in A$ such that $f(x)=y\text{.}$

3.

Let $g:\mathbb{R}\to \mathbb{R}$ be defined by $g(x)=5x+3\text{,}$ for all $x\in \mathbb{R}\text{.}$ Complete the following proofs of the following propositions about the function $g\text{.}$

Propostion 1

For all $a,b\in \mathbb{R}\text{,}$ if $g(a)=g(b)\text{,}$ then $a=b\text{.}$

Proof

We let $a,b\in \mathbb{R}\text{,}$ and we assume that $g(a)=g(b)$ and will prove that $a=b\text{.}$ Since $g(a)=g(b)\text{,}$ we know that

$$5a+3=5b+3\text{.}$$

(Now prove that in this situation, $a=b\text{.}$)

Proposition 2

For all $b\in \mathbb{R}\text{,}$ there exists an $a\in \mathbb{R}$ such that $g(a)=b\text{.}$

Proof

We let $b\in \mathbb{R}\text{.}$ We will prove that there exists an $a\in \mathbb{R}$ such that $g(a)=b$ by constructing such an $a$ in $\mathbb{R}\text{.}$ In order for this to happen, we need $g(a)=5a+3=b\text{.}$

(Now solve the equation for $a$ and then show that for this real number $a\text{,}$ $g(a)=b\text{.}$)