Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Let [tex]\( P \)[/tex] be the set of integers that are multiples of 3 between 1 and 20. Let [tex]\( Q \)[/tex] be the set of even natural numbers up to 15. Which of the following sets represents [tex]\( P \cap Q \)[/tex]?

Select the correct answer below:
A. [tex]\(\{6,12\}\)[/tex]
B. [tex]\(\{3,6,12\}\)[/tex]
C. [tex]\(\{1,6,12\}\)[/tex]
D. [tex]\(\{3,6\}\)[/tex]


Sagot :

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]