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.