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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 $\mathbb N^+$?

2 Answers 2

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$\mathbb R^+$ 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. $\mathbb N^+$.

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    @user477343: I don't understand you, and your notation is not consistent. So I'm going to stop replying now. Have a great day!2018-01-26
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Simply $\mathbb R$ means the set of real numbers.

$\mathbb R^+$ means the set of positive real numbers.

And $\mathbb R^-$ means the set of negative real numbers.