Let's begin by defining the sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].
1. Set [tex]\( P \)[/tex]: This is the set of integers that are multiples of 3 between 1 and 20. The multiples of 3 within this range are:
[tex]\[ 3, 6, 9, 12, 15, 18 \][/tex]
So, [tex]\( P = \{3, 6, 9, 12, 15, 18\} \)[/tex].
2. Set [tex]\( Q \)[/tex]: This is the set of even natural numbers up to 15. The even natural numbers within this range are:
[tex]\[ 2, 4, 6, 8, 10, 12, 14 \][/tex]
So, [tex]\( Q = \{2, 4, 6, 8, 10, 12, 14\} \)[/tex].
3. Intersection of Sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]: To find the intersection [tex]\( P \cap Q \)[/tex], we need the elements that are common to both sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].
From the sets [tex]\( P = \{3, 6, 9, 12, 15, 18\} \)[/tex] and [tex]\( Q = \{2, 4, 6, 8, 10, 12, 14\} \)[/tex], the common elements are:
[tex]\[ 6 \][/tex] and [tex]\[ 12 \][/tex]
Thus, the intersection [tex]\( P \cap Q \)[/tex] is:
[tex]\[ P \cap Q = \{6, 12\} \][/tex]
Given this detailed breakdown, the correct answer is:
[tex]\[ \{6,12\} \][/tex]
So, the answer is the first option:
[tex]\[ \boxed{\{6,12\}} \][/tex]
