Let A = { a,b, {a,c},Ø} then find

a) A-{a,c}

b) A- {{a,b}}

c) Ø-A d) {a,c}-A

Given the set \( A = \{ a, b, \{a, c\}, \emptyset \} \), we will solve each part of the problem:

### Part (a): \( A - \{a, c\} \)

The set \( \{a, c\} \) is an element of \( A \), so we need to remove it from \( A \):

\[ A = \{ a, b, \{a, c\}, \emptyset \} \]

Removing \( \{a, c\} \) from \( A \):

\[ A - \{a, c\} = \{ a, b, \{a, c\}, \emptyset \} - \{ \{a, c\} \} = \{ a, b, \emptyset \} \]

### Part (b): \( A - \{ \{a, b\} \} \)

The set \( \{a, b\} \) is not an element of \( A \), so \( A \) remains unchanged:

\[ A - \{ \{a, b\} \} = \{ a, b, \{a, c\}, \emptyset \} \]

### Part (c): \( \emptyset - A \)

The set difference of any set \( B \) with the empty set \( \emptyset \) is always \( \emptyset \):

\[ \emptyset - A = \emptyset \]

### Part (d): \( \{a, c\} - A \)

Here, \( \{a, c\} \) is not an element but a subset of \( A \). We find the difference by removing the elements of \( A \) that are in \( \{a, c\} \):

Since neither \( a \) nor \( c \) are elements of \( A \):

\[ \{a, c\} - A = \{a, c\} \]

### Summary of Results:

a) \( A - \{a, c\} = \{ a, b, \emptyset \} \)

b) \( A - \{ \{a, b\} \} = \{ a, b, \{a, c\}, \emptyset \} \)

c) \( \emptyset - A = \emptyset \)

d) \( \{a, c\} - A = \{a, c\} \)

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Do you want any further details or have any questions?

Here are some additional questions to expand on this topic:

1. What is the result of \( A - \{b\} \)?
2. How would you describe the difference between set difference and subset?
3. What is the union of \( A \) and \( \{d, e\} \)?
4. If \( B = \{a, b\} \), what is \( A \cup B \)?
5. What is the intersection of \( A \) and \( \{a, \{a, c\}\} \)?
6. How does the presence of \( \emptyset \) affect set operations?
7. What is the complement of \( A \) if the universal set is \( \{a, b, c, d, \{a, c\}, \emptyset\} \)?
8. How do nested sets like \( \{a, c\} \) complicate set operations?

**Tip:** When performing set operations, always check if the elements involved are subsets or individual elements to avoid confusion.

Explore set operations with examples including set differences, unions, and intersections. Learn how to apply set theory concepts with detailed solutions to various problems.

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