Insert ∈ or ∉ in each blank to make the resulting statement true. ∅ ____ ∅

Verified step by step guidance

1. Understand the symbols involved: ∅ represents the empty set, which contains no elements.
2. Recall the meaning of the symbols ∈ and ∉: ∈ means 'is an element of', and ∉ means 'is not an element of'.
3. Analyze the statement ∅ ____ ∅: The first ∅ is the object we are checking, and the second ∅ is the set we are checking membership in.
4. Since the empty set ∅ is a set itself, it is not an element of the empty set (which has no elements). Therefore, the correct symbol to use is ∉.
5. So, the completed statement is ∅ ∉ ∅, meaning the empty set is not an element of the empty set.

Key Concepts

Empty Set (∅)

The empty set, denoted by ∅, is the set containing no elements. It is unique and is a subset of every set, including itself. Understanding the properties of the empty set is fundamental in set theory and algebra.

Set Membership (∈ and ∉)

The symbol ∈ means 'is an element of,' indicating membership of an element in a set. Conversely, ∉ means 'is not an element of.' These symbols are used to express whether a specific object belongs to a given set.

Difference Between Element and Subset

An element is a single object contained in a set, while a subset is a set whose elements all belong to another set. The empty set is a subset of every set but is not an element of itself, which is crucial for correctly using ∈ or ∉.