Proving $g: \mathbb{Z} \to \mathbb{R}$, $g(x)=2x+3$ is one to one.

Question

Working on the book: Daniel J. Velleman. "HOW TO PROVE IT: A Structured Approach, Second Edition" (p. 242)

2. We can define a function $g: \mathbb{Z} \to \mathbb{R}$ by the rule that for every $x \in \mathbb{Z}$, $g(x) = 2x + 3$.

• Assume $a,a' \in \mathbb{Z} \land g(a)=g(a')$
  • $g(a)=g(a')$
  • $2a+3=2a'+3$
  • $2a=2a'$
  • ...

I know $a \in \mathbb{Z}$ and division is not closed in $\mathbb{Z}$. Now, my question is: how can I justify $a=a'$ ?

Answer 1

You know what set you're working in by knowing $g(a), g(a') \in \mathbb{R}$. Therefore performing the following computation in $\mathbb{R}$ allows you to deduce $a = a'$ from $2a = 2a'$ since $\mathbb{R}$ is a field.