Let’s analyze and solve each problem step-by-step. The symbols here are related to set theory:

1. ϕ (phi) represents the empty set.
2. {ϕ} represents the set containing the empty set (a single-element set where that element is the empty set).

Solving the problems:

(a) $\emptyset \cap \{\emptyset\}$

• Explanation: The intersection of the empty set ($\emptyset$) with $\{\emptyset\}$ is the set of elements they share.
  • Since $\emptyset$ is empty, there are no elements to share.
• Result: $\emptyset$

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

• Explanation:
  • $\{\emptyset, \{\emptyset\}\}$ is a set containing two elements: $\emptyset$ and $\{\emptyset\}$.
  • Subtracting $\{\{\emptyset\}\}$ means removing $\{\emptyset\}$ (the set containing the empty set) from the original set.
  • The remaining element is $\emptyset$.
• Result: $\{\emptyset\}$

(c) $\{\emptyset, \{\emptyset\}\} - \{\emptyset\}$

• Explanation:
  • Similar to (b), but this time we are subtracting $\{\emptyset\}$, which removes the empty set $\emptyset$ from $\{\emptyset, \{\emptyset\}\}$.
  • The remaining element is $\{\emptyset\}$.
• Result: $\{\{\emptyset\}\}$

(d) $\{\{\{\emptyset\}\}\} - \emptyset$

• Explanation:
  • $\{\{\{\emptyset\}\}\}$ is a set containing one element: $\{\{\emptyset\}\}$.
  • Subtracting $\emptyset$ does not remove anything because $\emptyset$ is not an element of $\{\{\{\emptyset\}\}\}$.
• Result: $\{\{\{\emptyset\}\}\}$

Final Answers:

1. (a) $\emptyset$
2. (b) $\{\emptyset\}$
3. (c) $\{\{\emptyset\}\}$
4. (d) $\{\{\{\emptyset\}\}\}$

Let me know if you’d like a deeper explanation or additional examples! 😊

Related Questions:

1. What is the difference between $\emptyset$ and $\{\emptyset\}$?
2. How does subtraction work in set theory?
3. Can $\{\emptyset, \{\emptyset\}\}$ be a subset of another set?
4. What happens when you subtract $\emptyset$ from any set?
5. How would the operation $\{\emptyset\} \cap \{\{\emptyset\}\}$ be solved?

Tip:

Always distinguish between an empty set ($\emptyset$) and a set containing the empty set ($\{\emptyset\}$); they are entirely different!