Let's go through this step-by-step to solve the given problem.
### Part I: Venn Diagram Illustration
To illustrate the information accurately on a Venn diagram, let’s define the sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
- Set [tex]\( P \)[/tex]: Multiples of 3 less than 24. So, [tex]\( P = \{3, 6, 9, 12, 15, 18, 21\} \)[/tex].
- Set [tex]\( Q \)[/tex]: Even numbers from 1 to 20. So, [tex]\( Q = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \)[/tex].
We can represent these sets on a Venn diagram. Draw two intersecting circles, one for [tex]\( P \)[/tex] and one for [tex]\( Q \)[/tex]. Place the elements of each set in the appropriate regions. The intersection part will contain elements that are common to both sets.
### Part II: Calculations
1. Finding [tex]\( P \cup Q \)[/tex]:
- Union of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]: This is the set of all elements that are in [tex]\( P \)[/tex] or [tex]\( Q \)[/tex] or both.
[tex]\[
P \cup Q = \{2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21\}
\][/tex]
2. Finding [tex]\( P \cap Q \)[/tex]:
- Intersection of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]: This is the set of all elements that are both in [tex]\( P \)[/tex] and [tex]\( Q \)[/tex].
[tex]\[
P \cap Q = \{6, 12, 18\}
\][/tex]
3. Cardinality of Sets:
- [tex]\( n(Q) \)[/tex]: Number of elements in set [tex]\( Q \)[/tex].
[tex]\[
n(Q) = 10
\][/tex]
- [tex]\( n(P) \)[/tex]: Number of elements in set [tex]\( P \)[/tex].
[tex]\[
n(P) = 7
\][/tex]
- [tex]\( n(P \cup Q) \)[/tex]: Number of elements in set [tex]\( P \cup Q \)[/tex].
[tex]\[
n(P \cup Q) = 14
\][/tex]
### Summary:
- Set [tex]\( P \)[/tex]: [tex]\(\{3, 6, 9, 12, 15, 18, 21\}\)[/tex]
- Set [tex]\( Q \)[/tex]: [tex]\(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}\)[/tex]
- Union [tex]\( P \cup Q \)[/tex]: [tex]\(\{2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21\}\)[/tex]
- Intersection [tex]\( P \cap Q \)[/tex]: [tex]\(\{6, 12, 18\}\)[/tex]
### Number of Elements:
- [tex]\( n(Q) = 10 \)[/tex]
- [tex]\( n(P) = 7 \)[/tex]
- [tex]\( n(P \cup Q) = 14 \)[/tex]
You can draw a Venn diagram with circles for [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] where they intersect at [tex]\(\{6, 12, 18\}\)[/tex] to visualize these sets accurately.
