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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 $\mathbb N^+$?

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up vote 11 down vote accepted

$\mathbb R^+$ 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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I see now. I thought it meant that R+ included some extra element. This interpretation makes much more sense. – James Aug 26 '12 at 20:02
Note that $\mathbb N^+ = \mathbb Z^+$. Also if you want to confuse your readers, you can write the empty set as $\mathbb N^-$, the set of negative natural numbers. :-) – celtschk Aug 26 '12 at 20:40
NB : Depending on the country, $\mathbb R^+$ is also used for the set of non-negative real numbers. – Student Aug 26 '12 at 22:15
@Student: Really? $0\in\mathbb R^+$? Sounds bizarre! – Asaf Karagila Aug 26 '12 at 22:38
@Carl: Read the first line, I also say it defines the positive numbers. I just never thought zero is positive... :-) (Also, these French just have to do everything the other way around...! :-)) – Asaf Karagila Aug 27 '12 at 1:19

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.

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