To find the intersection of [tex]\( S \)[/tex] with the union of sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex], we need to perform the following steps:
1. Calculate the union of sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
The union [tex]\( P \cup Q \)[/tex] is the set containing all unique elements from both [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].
Given:
[tex]\[
P = \{6, 7, 11, 12, 15\}
\][/tex]
[tex]\[
Q = \{4, 7, 12, 15, 20\}
\][/tex]
The union [tex]\( P \cup Q \)[/tex] is:
[tex]\[
\{4, 6, 7, 11, 12, 15, 20\}
\][/tex]
2. Calculate the intersection of set [tex]\( S \)[/tex] with the union [tex]\( P \cup Q \)[/tex]:
The intersection [tex]\( S \cap (P \cup Q) \)[/tex] is the set of elements that are common to both [tex]\( S \)[/tex] and [tex]\( P \cup Q \)[/tex].
Given:
[tex]\[
S = \{3, 4, 11, 12, 16\}
\][/tex]
Using the union [tex]\( P \cup Q = \{4, 6, 7, 11, 12, 15, 20\} \)[/tex], we find the common elements with [tex]\( S \)[/tex]:
[tex]\[
S = \{3, 4, 11, 12, 16\}
\][/tex]
The intersection [tex]\( S \cap (P \cup Q) \)[/tex]:
[tex]\[
\{4, 11, 12\}
\][/tex]
Thus, the set that represents [tex]\( S \cap (P \cup Q) \)[/tex] is [tex]\( \{4, 11, 12\} \)[/tex].
The correct answer is:
[tex]\[
\{4, 11, 12\}
\][/tex]
