Math Problem Statement

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Solution

The expression \emptyset \subseteq \emptyset (read as "the empty set is a subset of the empty set") is true.

Here's why:

  • The empty set \emptyset is a set that contains no elements.
  • A set AA is considered a subset of another set BB (written ABA \subseteq B) if every element of AA is also an element of BB.
  • Since \emptyset has no elements, the condition "every element of \emptyset is also an element of \emptyset" is vacuously true. Therefore, \emptyset is a subset of itself.

Thus, \emptyset \subseteq \emptyset is indeed true.

Would you like more details or have any questions?

Here are five related questions you might consider:

  1. What is the difference between \subset and \subseteq?
  2. Can a set be a subset of itself if it is not empty?
  3. What is the power set of the empty set?
  4. How do you prove that any set is a subset of itself?
  5. What does it mean for one set to be a proper subset of another?

Tip: When working with set theory, always remember that an empty set is a subset of every set, including itself.