# why is $(0,1) \subseteq$ $\mathbb{R}$ \ $\mathbb{N}$

why is $(0,1) \subseteq$ $\mathbb{R}$ \ $\mathbb{N}$

Sorry it seems very simple but can't get my mind to understand why, I feel like $\mathbb{R}$ \ $\mathbb{N}$ = {all negative numbers and irrational numbers }

doesn't the $\subseteq$ mean that all elements in one set is contained in the other?

• $\mathbb{N}$ means natural number, rational is usually denoted by $\mathbb{Q}$. Jun 8, 2015 at 3:27
• $\mathbb{N}$ is the natural numbers, not the rational numbers. That is, $\mathbb{N}=\{0,1,2,\dots\}$. Jun 8, 2015 at 3:27
• @NoahOlander You have a weird definition for $\Bbb N$. Everyone knows $\Bbb N = \{1,2,3,\ldots\}$. :P Jun 8, 2015 at 3:28
• You mean $\{ 0 ,1 \}$, the set of $0$ and $1$? If this is the case, then the claim is false; for $\{ 1 \} \subset \mathbb{N}$. Jun 8, 2015 at 3:29
• @JMoravitz I was just making a really dry joke. It's a fun harmless argument to have. Jun 8, 2015 at 3:38

Yes, $A\subseteq B$ means that each element of $A$ is also an element of $B$. And that is true if $A=(0,1)$ and $B=\Bbb R\setminus\Bbb N$. First, it should be clear that every $x\in(0,1)$ is a real number, so $(0,1)\subseteq\Bbb R$. Does $(0,1)$ contain any natural number? No: it contains only the numbers strictly between $0$ and $1$, which don’t even include any integers, let alone natural numbers. Thus, $x\notin\Bbb N$ whenever $x\in(0,1)$. Put the pieces together: if $x\in(0,1)$, then $x\in\Bbb R$, and $x\notin\Bbb N$, so $x\in\Bbb R\setminus\Bbb N$. Thus, $(0,1)\subseteq\Bbb R\setminus\Bbb N$.

• thanks! that makes sense i guess for some reason the (0,1) didnt register as not including 0 and 1 as an element contained. Jun 8, 2015 at 4:00
• @Jared: You’re welcome! I wondered if that might be the problem. Jun 8, 2015 at 4:03

No,no it means that $\mathbb R$ represents the set of real numbers and $\mathbb N$ represents the set of natural numbers. as usual, and your definition of subset is correct but i think you should remove $\mathbb N$ and only represent it using Real numbers.because it would include $0+\Delta(t)$,where $\Delta(t)$ represents really small, infinitesimal number. In decimals may be rational or irrational,which are both contained in real numbers. and you know my friend that $\mathbb N \subseteq [1,\infty)$,which can also be understood as a separate constraint for not taking natural numbers as our criteria.

hope you understand it!