Let's analyze and solve the set operations given in the image. The problem defines two sets:

• $A = \{ a, b, \{ a, c \}, \emptyset \}$
• $B = \{ a, \{ a \}, d, e \}$

We need to find the results of several set operations:

(a) $A - \emptyset$

Subtracting the empty set from $A$ does not change $A$, so: $A - \emptyset = \{ a, b, \{ a, c \}, \emptyset \}$

(b) $A - \{\emptyset\}$

This operation removes the empty set from $A$: $A - \{\emptyset\} = \{ a, b, \{ a, c \} \}$

(c) $\{\{ a, c \}\} - A$

Here, the set $\{\{a, c\}\}$ is a set containing the set $\{a, c\}$. Since $\{a, c\}$ is already in $A$, the result is: $\{\{ a, c \}\} - A = \emptyset$

(d) $A \oplus B$ (Symmetric Difference)

The symmetric difference includes elements in either $A$ or $B$, but not in both: $A \oplus B = \{ b, \{ a, c \}, \emptyset, \{ a \}, d, e \}$

(e) $\{a\} - \{A\}$

The set $\{a\}$ is just {a}, and $\{A\} = \{\{a, b, \{a, c\}, \emptyset\}\}$. There is no common element, so: $\{a\} - \{A\} = \{ a \}$

(f) $P(A - B)$ (Power Set of $A - B$)

First, calculate $A - B$: $A - B = \{ b, \{ a, c \}, \emptyset \}$ The power set $P(A - B)$ includes all subsets: $P(A - B) = \{\emptyset, \{b\}, \{\{a, c\}\}, \{\emptyset\}, \{b, \{a, c\}\}, \{b, \emptyset\}, \{\{a, c\}, \emptyset\}, \{b, \{a, c\}, \emptyset\}\}$

(g) $\emptyset - A$

The empty set minus any set is still the empty set: $\emptyset - A = \emptyset$

(h) $B^2$ (Cartesian Product of $B$ with itself)

The Cartesian product $B \times B$: $B^2 = \{ (a, a), (a, \{a\}), (a, d), (a, e), (\{a\}, a), (\{a\}, \{a\}), (\{a\}, d), (\{a\}, e), (d, a), (d, \{a\}), (d, d), (d, e), (e, a), (e, \{a\}), (e, d), (e, e) \}$

(i) $A \cup (B \cap A)$

First, find the intersection $B \cap A$: $B \cap A = \{ a \}$ Then the union with $A$: $A \cup (B \cap A) = \{ a, b, \{ a, c \}, \emptyset \}$

(j) $A \cap P(A)$

The power set $P(A)$ includes all subsets of $A$. Since $A$ itself is not a subset of $P(A)$, the intersection is: $A \cap P(A) = \emptyset$

Would you like any more details on these solutions or further clarifications?

Related Questions:

1. How do you compute the power set of a given set?
2. What are the properties of symmetric difference in set theory?
3. How is the Cartesian product of sets defined, and what are its applications?
4. Can you explain more about set operations like union, intersection, and difference?
5. How do nested sets affect set operations?

Tip:

Always carefully distinguish between elements and subsets when performing set operations, as confusing these can lead to incorrect results!