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One of the math problems I have describes a set of numbers this way:

let there be a set A such that $A=\{{1,2,3,...,n+3}\}$.

I don't understand what the $n+3$ means and how the set actually looks like.

Caleb Stanford
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Yos
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5 Answers5

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The set is all positive integers between $1$ and $n + 3$, inclusive.

If $n = 4$, for example, the set is $$ \{1, 2, 3, 4, 5, 6, 7\}. $$ And if $n = 2$, then it's $$ \{1,2,3,4,5\}. $$

It may also be helpful to include a few more terms: $\{1, 2, 3, 4, \ldots, n, n+1, n+2, n+3\}$.

In general, $\{1, 2, 3, \ldots k\}$ is shorthand for the set of positive integers between $1$ and $k$, inclusive.

Caleb Stanford
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  • thank you. Now I understand but it seems counter intuitive to define a set this way, would be much simpler to just say $1,2,3...n$ – Yos Sep 20 '16 at 17:53
  • @Yos But if $n$ is defined already, then that would be a different set. That set has the positive integers between $1$ and $n$, but we want the positive integers between $1$ and $n + 3$. – Caleb Stanford Sep 20 '16 at 17:55
  • I guess that makes sense, just haven't seen such definitions before. – Yos Sep 20 '16 at 18:02
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The $n$ is arbitrary and finite. Basically, choose an $n$ so

$$ A_n = \{ 1, 2, \ldots, n, n+1, n+2, n+3 \}.$$

For example, $A_4 = \{1, 2, \ldots, 5, 6, 7 \}.$

Sean Roberson
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$n$ is just a natural number. So the set is of variable length. For example choosing $n=2$ yields $$A =\{1,2,3,4,5\}$$

TheGeekGreek
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For this to have a meaning, $n$ must be already defined.

For instance if $n=3$,

$$A=\{1,2,\ldots,n+3\}=\{1,2,3,4,5,6\}.$$

The "$n+3$" means that you are enumerating all integers from $1$ to $n+3$.

Another notation (to avoid the dots) would be

$$A=\{i\}_{1\leq i\leq n+3}.$$

E. Joseph
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Assuming for instance that $n$ must be a positive integer, this would be a way of forcing the set to have at least $4$ elements.

paw88789
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