If P(n) is a set prime factors of n evaluate P(42) U P(45), P(42) U P(42) - P(36).

The universal set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
The given subsets are:
$X = \{2, 3, 4, 6\}$
$Y = \{1, 3, 4, 5, 8, 10\}$
$Z = \{1, 3, 5\}$

First, observe that all elements of $Z$ are also in $Y$, so $Z$ is a subset of $Y$ ($Z \subset Y$). This simplifies the Venn diagram structure, as the circle for $Z$ will be entirely inside the circle for $Y$.

Calculate intersections:
$X \cap Y = \{3, 4\}$
$X \cap Z = \{3\}$
$Y \cap Z = \{1, 3, 5\}$ (Since $Z \subset Y$, this is simply $Z$)
$X \cap Y \cap Z = \{3\}$

Calculate elements for each distinct region:

• Elements in $X \cap Y \cap Z$: $\{3\}$
• Elements in $X \cap Y$ but not in $Z$: $(X \cap Y) \setminus Z = \{3, 4\} \setminus \{1, 3, 5\} = \{4\}$
• Elements in $Y \cap Z$ but not in $X$: $(Y \cap Z) \setminus X = \{1, 3, 5\} \setminus \{2, 3, 4, 6\} = \{1, 5\}$
• Elements in $X$ only: $X \setminus (Y \cup Z) = \{2, 3, 4, 6\} \setminus (\{1, 3, 4, 5, 8, 10\} \cup \{1, 3, 5\}) = \{2, 3, 4, 6\} \setminus \{1, 3, 4, 5, 8, 10\} = \{2, 6\}$
• Elements in $Y$ only (outside $X$ and $Z$): $Y \setminus (X \cup Z) = \{1, 3, 4, 5, 8, 10\} \setminus (\{2, 3, 4, 6\} \cup \{1, 3, 5\}) = \{1, 3, 4, 5, 8, 10\} \setminus \{1, 2, 3, 4, 5, 6\} = \{8, 10\}$
• Elements in $Z$ only: $Z \setminus (X \cup Y) = \emptyset$ (as $Z \subset Y$, and all elements are accounted for in $X \cap Y \cap Z$ or $(Y \cap Z) \setminus X$)
• Elements outside $X \cup Y \cup Z$: $U \setminus (X \cup Y \cup Z) = \{1, ..., 10\} \setminus (\{1, 2, 3, 4, 5, 6, 8, 10\}) = \{7, 9\}$

A Venn diagram would show three overlapping circles for $X$, $Y$, and $Z$. The circle for $Z$ would be drawn entirely inside the circle for $Y$. The elements are placed in the corresponding regions:

• The region common to $X$, $Y$, and $Z$ (i.e., $X \cap Z$): $\{3\}$
• The region common to $X$ and $Y$ but outside $Z$: $\{4\}$
• The region common to $Y$ and $Z$ but outside $X$: $\{1, 5\}$
• The region for $X$ only: $\{2, 6\}$
• The region for $Y$ only (outside $X$ and $Z$): $\{8, 10\}$
• The region outside all three sets: $\{7, 9\}$

First, find the complement of $X$, denoted $X'$.
$X' = U \setminus X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{2, 3, 4, 6\} = \{1, 5, 7, 8, 9, 10\}$
Now, find the intersection of $X'$ and $Y$:
$X' \cap Y = \{1, 5, 7, 8, 9, 10\} \cap \{1, 3, 4, 5, 8, 10\}$

First, find the complement of $Z$, denoted $Z'$.
$Z' = U \setminus Z = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \setminus \{1, 3, 5\} = \{2, 4, 6, 7, 8, 9, 10\}$
Next, find the intersection of $X$ and $Y$:
$X \cap Y = \{2, 3, 4, 6\} \cap \{1, 3, 4, 5, 8, 10\} = \{3, 4\}$
Finally, find the union of $Z'$ and $(X \cap Y)$:
$Z' \cup (X \cap Y) = \{2, 4, 6, 7, 8, 9, 10\} \cup \{3, 4\}$