{1,2,3} ⊆ {1,2,3,4,5}

{1,2,3} ⊂ {1,2,3,4,5}

Is ∅ ⊆ {1,2,3}?

Yes! ∀x (x ∈ ∅) → (x ∈ {1,2,3}) holds (for all

over empty domain)

Does ∅ ∈ {1,2,3}?

No!

Is ∅ ⊆ {∅,1,2,3}?

Yes!

Does ∅ ∈ {∅,1,2,3}?

Yes!  but how plz explain ??

Let's break this down step by step to explain the concept of sets, subsets, and membership in more detail.

### Definitions:
1. **Subsets (⊆)**: A set \( A \) is a subset of a set \( B \) (denoted \( A \subseteq B \)) if every element of \( A \) is also an element of \( B \). It's possible for \( A \) and \( B \) to be equal.
2. **Proper Subsets (⊂)**: A set \( A \) is a proper subset of a set \( B \) (denoted \( A \subset B \)) if \( A \) is a subset of \( B \), but \( A \neq B \) (i.e., \( A \) has fewer elements than \( B \)).
3. **Empty Set (∅)**: The empty set is a set that contains no elements. 

### Statements from your example:
1. **Is ∅ ⊆ {1, 2, 3}?**  
   **Yes**, because the empty set is a subset of every set, including {1, 2, 3}. By definition, there are no elements in ∅, so there’s no counterexample to challenge this statement.

2. **Does ∅ ∈ {1, 2, 3}?**  
   **No**, because the set {1, 2, 3} contains the elements 1, 2, and 3, but not the empty set ∅ as one of its elements.

3. **Is ∅ ⊆ {∅, 1, 2, 3}?**  
   **Yes**, for the same reason as above. The empty set is a subset of every set, including the set {∅, 1, 2, 3}.

4. **Does ∅ ∈ {∅, 1, 2, 3}?**  
   **Yes**, because in this case, the set {∅, 1, 2, 3} contains the empty set ∅ as one of its elements, alongside 1, 2, and 3. This is different from the previous example, where ∅ was not one of the elements of the set {1, 2, 3}. 

### Key Concepts:
- **Membership (∈)** refers to whether an element is part of a set. In the statement "Does ∅ ∈ {∅, 1, 2, 3}?", you're asking whether the set ∅ itself is one of the elements of the set {∅, 1, 2, 3}. Since {∅, 1, 2, 3} explicitly includes ∅ as one of its members, the answer is yes.
  
- **Subset (⊆)** refers to whether all elements of one set are contained within another. The empty set is always a subset of any set, even of itself, because it has no elements that could contradict the subset condition.

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### Related Questions:
1. What is the difference between \( \subseteq \) and \( \subset \)?
2. Can a set be both an element and a subset of another set?
3. Why is the empty set considered a subset of every set?
4. How does the definition of subset apply to infinite sets?
5. What are some examples of sets where \( \emptyset \in A \) but \( \emptyset \not\subseteq A \)?

### Tip:
Always remember, a set being an *element* of another set is different from it being a *subset*. Membership means the set itself is contained as an individual entity, while a subset means all elements of one set are included in the other.

The query submitted by the user: {1,2,3} ⊆ {1,2,3,4,5}

{1,2,3} ⊂ {1,2,3,4,5}

Is ∅ ⊆ {1,2,3}?

Yes! ∀x (x ∈ ∅) → (x ∈ {1,2,3}) holds (for all over empty domain)

Does ∅ ∈ {1,2,3}?

No!

Is ∅ ⊆ {∅,1,2,3}?

Yes!

Does ∅ ∈ {∅,1,2,3}?

Yes!  but how plz explain ??

Explore key concepts of set theory including subsets, proper subsets, and membership. Learn why the empty set is always a subset of any set, and understand the difference between an element being a member of a set versus a subset. Suitable for high school students in Grades 9-12.

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