For a function $f$ that maps set $A$ to $B$,
- $f\colon\mathbb R^+\to\mathbb R^+$, $f(x) = x^2$ is injective.
- $f\colon\mathbb R\to\mathbb R$, $f(x) = x^2$ is not injective since $(- x)^2 = x^2$.
what is the difference between $\mathbb R^+$ and $\mathbb R$?
Additionally, what is the difference between $\mathbb N$ and $\mathbb N^+$?
Best Answer
$\mathbb R^+$ commonly denotes the set of positive real numbers, that is: $$\mathbb R^+ = \{x\in\mathbb R\mid x>0\}$$
It is also denoted by $\mathbb R^{>0},\mathbb R_+$ and so on.
For $\mathbb N$ and $\mathbb N^+$ the difference is similar, however it may be non-existent if you define $0\notin\mathbb N$. In many set theory books $0$ is a natural number, while in analysis it is often not considered a natural number. Your mileage may vary on $\mathbb N$ vs. $\mathbb N^+$.