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Since natural numbers($\mathbb{N}$) are set of non-negative integers ... Does it mean that positive integer can be a subset of natural numbers?

For example:

Is {2,3,4} a subset of $\mathbb{N}$?

Thanks.

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Is zero a Natural number? –  Hassan Muhammad Nov 3 '11 at 9:32
    
Yes. Zero is a natural number –  xscape Nov 25 '11 at 10:57

2 Answers 2

up vote 2 down vote accepted

Definition: Let $A,B$ be sets. We say that $A$ is a subset of $B$, if for every $a\in A$ it holds that $a\in B$. If this is the case we write $A\subseteq B$.

Now consider your sets, $A=\{2,3,4\}$ and $B=\{0,1,2,3,4,5\ldots\}=\mathbb N$. For every element in $A$ we have that it is a positive integer, and it is in $B$.

By the definition of inclusion, $A\subseteq B$.

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If your definition of $\mathbb{N}$ is non-negative integer, then $\mathbb{N}$ is the set $\{0,1,2,3,...\}$. So to answer your question, yes $\{2,3,4,...\}$ is a subset of $\mathbb{N}$ because every element in $\{2,3,4,...\}$ is a non-negative integer.

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Although the question asked about $\lbrace2,3,4\rbrace$, not $\lbrace2,3,4,\dots\rbrace$. Of course, the answer is still "yes". –  Gerry Myerson Nov 3 '11 at 5:32

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