Insert ⊆ or ⊈ in each blank to make the resulting statement true. ∅ ____ {1, 4, 6, 8}

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Recall the definitions: The symbol \( \subseteq \) means 'is a subset of,' which includes the possibility of being equal, while \( s \) (likely intended as \( \subset \)) means 'is a proper subset of,' which excludes equality.

Identify the sets involved: The empty set \( \emptyset \) and the set \( \{1, 4, 6, 8\} \).

Understand that the empty set \( \emptyset \) has no elements, so it is a subset of every set, including \( \{1, 4, 6, 8\} \).

Since \( \emptyset \) is not equal to \( \{1, 4, 6, 8\} \), it is also a proper subset of \( \{1, 4, 6, 8\} \).

Therefore, the correct symbol to fill in the blank is the proper subset symbol \( \subset \) (or 's' as given), making the statement \( \emptyset \subset \{1, 4, 6, 8\} \) true.

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Key Concepts

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

Empty Set (∅)

The empty set, denoted by ∅, is the set containing no elements. It is a subset of every set because there are no elements in ∅ that could violate the subset condition. Understanding the empty set is fundamental when dealing with subset relations.

Subset Symbol (⊆)

The subset symbol ⊆ indicates that every element of the first set is also an element of the second set. If A ⊆ B, then all elements of A are contained in B. This includes the possibility that A and B are equal sets.

Proper Subset Symbol (⊂ or s)

A proper subset means that all elements of the first set are in the second set, but the two sets are not equal. The symbol s (often written as ⊂) is used to denote a proper subset. If A s B, then A ⊆ B and A ≠ B.

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