(c) Write $$ \left\{x \in U \mid x=n^{2}\right. $$ for some $$ \left.n \in N\right\} $$ by listing the elements in braces. Choose the correct answer below. A. $$ \{1,4,9,16\} $$ B. C. $$ \{1,2,3,4\} $$ D. $$ \{2,3,5,7,8,10,11,12,13,14,15,17,18,19,20\} $$

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UpStudy AI · Accepted Answer

We need to list the elements $$ x $$ in the universe $$ U $$ (which, based on the context, is likely the set $$\{1, 2, \dots, 20\}$$) such that $$ x $$ is a perfect square, that is, $$ x = n^2 $$ for some $$ n \in \mathbb{N} $$.

1. We start with $$ n = 1 $$:
   $$
   1^2 = 1
   $$
   Since $$ 1 \in U $$, include $$ 1 $$.

2. Next, for $$ n = 2 $$:
   $$
   2^2 = 4
   $$
   Since $$ 4 \in U $$, include $$ 4 $$.

3. For $$ n = 3 $$:
   $$
   3^2 = 9
   $$
   Since $$ 9 \in U $$, include $$ 9 $$.

4. For $$ n = 4 $$:
   $$
   4^2 = 16
   $$
   Since $$ 16 \in U $$, include $$ 16 $$.

5. For $$ n = 5 $$:
   $$
   5^2 = 25
   $$
   Given that $$ 25 $$ is not in $$ U $$ (as assumed $$ U = \{1,2,\ldots,20\} $$), we stop here.

Thus, the set is:
$$
\{1, 4, 9, 16\}
$$

Comparing with the provided options, the correct answer is:

A. $$ \{1,4,9,16\} $$
The correct answer is $$ \{1, 4, 9, 16\} $$, which is option A.

(c) Write \( \left\{x \in U \mid x=n^{2}\right. \) for some \( \left.n \in N\right\} \) by listing the elements in braces. Choose the correct answer below. A. \( \{1,4,9,16\} \) B. C. \( \{1,2,3,4\} \) D. \( \{2,3,5,7,8,10,11,12,13,14,15,17,18,19,20\} \)

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