We need the set of all squares of integers that are strictly between 1 and 1000. That is, we want

\[
\{ s \mid s=n^2,\, n\in \mathbb{Z},\ 1 < n^2 < 1000 \}.
\]

Since \(1<n^2\), note that if \(n=1\) then \(1^2=1\) (which is not allowed) and if \(n=0\) or \(n\) is negative, the square is either 0 or repeats a positive square already produced by a positive \(n\).

Thus, we consider only positive integers. We want the condition on \(n\) such that

\[
n^2 < 1000.
\]

Taking the square root (and knowing \(n\) is positive) we have

\[
n < \sqrt{1000}.
\]

Since \(31^2 = 961 < 1000\) and \(32^2 = 1024 > 1000\), we conclude that

\[
n \le 31.
\]

But recall the extra condition \(1<n^2\) excludes \(n=1\) (as \(1^2=1\) is not allowed). Therefore, \(n\) must satisfy

\[
1 < n \le 31.
\]

Expressing this in set builder notation for the squares from \(n=2\) to \(n=31\) gives

\[
\{ n^2 \mid n \in \mathbb{Z} \text{ and } 1<n<32 \}.
\]

Among the answer choices, this matches

\[
\{ n^2 \mid n \in \mathbb{Z} \text{ and } 1<n<32 \}.
\]

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Pregunta

Question
Use set builder notation to express the set of squares of integers between 1 and 1000 exclusively, i.e not including 1 and
1000 .
Select the correct answer below:
{ and
and
and
and

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The correct set builder notation is:

Solución

We need the set of all squares of integers that are strictly between 1 and 1000. That is, we want

Since

, note that if

then

(which is not allowed) and if

is negative, the square is either 0 or repeats a positive square already produced by a positive

Thus, we consider only positive integers. We want the condition on

such that

Taking the square root (and knowing

is positive) we have

Since

and

, we conclude that

But recall the extra condition

excludes

(as

is not allowed). Therefore,

must satisfy

Expressing this in set builder notation for the squares from

gives

Among the answer choices, this matches

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Como

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Simplifique esta solución

Mind Expander

The correct expression in set builder notation for the set of squares of integers between 1 and 1000 exclusively is

. This is because the squares of the integers from 2 to 31 (since

which is less than 1000) will yield the desired set of squares—specifically,

Now let’s move on to some fun tidbits! Did you know that the earliest known use of squares can be traced back to ancient Egypt, where they used geometric forms in their practical mathematics? It’s fascinating to think about how these concepts laid the groundwork for modern theories!

Also, if you’re looking to explore more about sets, consider diving into the world of functional notation and exploring how these foundational concepts are applied in computer science and data structures! It can be really eye-opening to see how abstract mathematical ideas find their way into programming and algorithms!

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