Let's take each part of the question step-by-step.
### Part (a):
List the elements of all sets:
1. Set [tex]\( U \)[/tex]:
[tex]\[
U = \{ x \mid x \leq 15, x \in \mathbb{N} \}
\][/tex]
This means [tex]\( U \)[/tex] consists of all natural numbers from 0 to 15:
[tex]\[
U = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 \}
\][/tex]
2. Set [tex]\( P \)[/tex]:
[tex]\[
P = \{ y \mid y \text{ is a square number}, y \in U \}
\][/tex]
Listing the square numbers in [tex]\( U \)[/tex]:
[tex]\[
P = \{ 0, 1, 4, 9 \}
\][/tex]
3. Set [tex]\( Q \)[/tex]:
[tex]\[
Q = \{ z \mid z \text{ is a composite number}, z \in U \}
\][/tex]
Listing the composite numbers in [tex]\( U \)[/tex]:
[tex]\[
Q = \{ 4, 6, 8, 9, 10, 12, 14, 15 \}
\][/tex]
### Part (b):
Find [tex]\( P \cup Q \)[/tex] and display in a Venn diagram:
1. Union of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
P \cup Q = \{ x \mid x \in P \text{ or } x \in Q \}
\][/tex]
Combining the elements from sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
P \cup Q = \{ 0, 1, 4, 6, 8, 9, 10, 12, 14, 15 \}
\][/tex]
Venn Diagram:
- Draw two intersecting circles: one for [tex]\( P \)[/tex] and one for [tex]\( Q \)[/tex].
- Place the elements [tex]\( \{ 4, 9 \} \)[/tex] in the intersection.
- Place [tex]\( \{ 0, 1 \} \)[/tex] only in [tex]\( P \)[/tex] (but not in [tex]\( Q \)[/tex]).
- Place [tex]\( \{ 6, 8, 10, 12, 14, 15 \} \)[/tex] only in [tex]\( Q \)[/tex] (but not in [tex]\( P \)[/tex]).
- Elements outside both sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex], but still in [tex]\( U \)[/tex], would be [tex]\( \{ 2, 3, 5, 7, 11, 13 \} \)[/tex].
### Part (c):
Intersection of [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
P \cap Q = \{ x \mid x \in P \text{ and } x \in Q \}
\][/tex]
Finding common elements between [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
[tex]\[
P \cap Q = \{ 4, 9 \}
\][/tex]
### Part (d):
Determining types of sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex]:
If [tex]\( P \cup Q \)[/tex] and [tex]\( P \cap Q \)[/tex] are equal, then the two sets would be equal. However, from the previous parts:
- [tex]\( P \cup Q = \{ 0, 1, 4, 6, 8, 9, 10, 12, 14, 15 \} \)[/tex]
- [tex]\( P \cap Q = \{ 4, 9 \} \)[/tex]
Since [tex]\( P \cup Q \neq P \cap Q \)[/tex], we conclude that the sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are not equal.
Thus, the sets [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are not equal.
