(c) Write \( \{x \in U \mid x=2 n+1 \) for some \( n \in N\} \) by listing the elements in braces. Choose the correct answer below. A. \( \varnothing \) B. \( \{3,5,7,9,11,13,15,17,19\} \) C. \( \{1,2,3,4,5,6,7,8,9\} \) D. \( \{2,4,6,8,10,12,14,16,18,20\} \)

To find the set \( \{x \in U \mid x=2n+1 \text{ for some } n \in \mathbb{N}\} \), we need to understand that this expression is generating all the odd numbers starting from 1 using \( n \) from the natural numbers \( \mathbb{N} \). Plugging in the values of \( n = 0, 1, 2, \ldots \): - For \( n = 0 \), \( x = 2(0) + 1 = 1 \) - For \( n = 1 \), \( x = 2(1) + 1 = 3 \) - For \( n = 2 \), \( x = 2(2) + 1 = 5 \) - For \( n = 3 \), \( x = 2(3) + 1 = 7 \) - For \( n = 4 \), \( x = 2(4) + 1 = 9 \) - For \( n = 5 \), \( x = 2(5) + 1 = 11 \) - ... Assuming \( U \) refers to the positive odd integers up to 20, the elements generated would be \( 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \). Looking at the provided options, B. \( \{3,5,7,9,11,13,15,17,19\} \) contains some but not all the generated elements and omits 1. The full odd set up to 19 isn't explicitly listed, so based on these details, none of the answers is entirely correct based on the standard interpretation of \( \mathbb{N} \). However, if we consider stopping at some maximum like \( 20 \) in \( U \), the resulting set (provided only the odd numbers up to 19 are included) would lead us to favor option B.