Question

let $u = \\{1,2,3, \dots 10\\}$, $a = \\{1,3,5,7\\}$, $b = \\{1,2,3,4\\}$, and $c = \\{3,4,6,7,9\\}$. select $(b \cup a) \cap (b \cup c)$ from the choices below. \bigcirc \\{1,2,3,5,10\\} \bigcirc \\{4,5,8,9,10\\} \bigcirc \\{3,6,8,9,10\\} \bigcirc \\{3,4,5,6,7\\} \bigcirc \\{1,2,3,4,7\\} \bigcirc \emptyset

Explanation:

Step1: Find \( B \cup A \)

The union of two sets \( B \) and \( A \) contains all elements that are in \( B \) or in \( A \) (or in both). Given \( A = \{1, 3, 5, 7\} \) and \( B = \{1, 2, 3, 4\} \), we combine the elements:
\( B \cup A=\{1, 2, 3, 4, 5, 7\} \)

Step2: Find \( B \cup C \)

The union of \( B \) and \( C \) contains all elements in \( B \) or \( C \) (or both). Given \( B = \{1, 2, 3, 4\} \) and \( C = \{3, 4, 6, 7, 9\} \), we combine the elements:
\( B \cup C=\{1, 2, 3, 4, 6, 7, 9\} \)

Step3: Find \( (B \cup A) \cap (B \cup C) \)

The intersection of two sets contains elements common to both sets. We find common elements between \( B \cup A = \{1, 2, 3, 4, 5, 7\} \) and \( B \cup C = \{1, 2, 3, 4, 6, 7, 9\} \):
\( (B \cup A) \cap (B \cup C)=\{1, 2, 3, 4, 7\} \)

Answer:

\(\{1,2,3,4,7\}\) (the last non - empty option among the choices, which is the option with set \(\{1,2,3,4,7\}\))