Insert ⊆ or ⊈ in each blank to make the resulting statement true. ∅ ____ ∅

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Recall the definitions of the symbols: The symbol \( \subseteq \) means 'is a subset of or equal to,' and the symbol \( \subset \) means 'is a proper subset of,' which excludes equality.

Consider the empty set \( \emptyset \). By definition, the empty set is a subset of every set, including itself.

Since \( \emptyset \) is equal to \( \emptyset \), the correct symbol to use to make the statement true is \( \subseteq \), because it allows for equality.

If you were to use \( \subset \), it would mean \( \emptyset \) is a proper subset of \( \emptyset \), which is false because a set cannot be a proper subset of itself.

Therefore, the correct statement is \( \emptyset \subseteq \emptyset \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Subset (⊆) Relation

A set A is a subset of set B (written A ⊆ B) if every element of A is also an element of B. This relation includes the possibility that A and B are equal sets. For example, the empty set ∅ is a subset of any set, including itself.

Proper Subset (⊂) Relation

A set A is a proper subset of set B (written A ⊂ B) if every element of A is in B, but A is not equal to B. This means B has at least one element not in A. The empty set ∅ cannot be a proper subset of itself because they are equal.

Empty Set (∅) Properties

The empty set ∅ is the unique set containing no elements. It is a subset of every set, including itself, but it is never a proper subset of itself. Understanding these properties helps determine which subset symbol correctly completes statements involving ∅.

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