Determine whether each statement is true or false. ∅ ∩ ∅ = ∅

Verified step by step guidance

1. Recall the definition of the intersection of two sets: \(A \cap B\) is the set of all elements that are in both \(A\) and \(B\).
2. Identify the sets involved: here both sets are the empty set \(\emptyset\), which contains no elements.
3. Apply the definition of intersection to \(\emptyset \cap \emptyset\): since there are no elements in either set, there are no elements common to both.
4. Therefore, the intersection \(\emptyset \cap \emptyset\) is also the empty set \(\emptyset\) because there are no shared elements.
5. Conclude that the statement \(\emptyset \cap \emptyset = \emptyset\) is true.

Key Concepts

Empty Set (∅)

The empty set, denoted by ∅, is the set that contains no elements. It is unique and serves as the foundation for set theory, representing the concept of 'nothing' or 'no members' within a set.

Set Intersection (∩)

The intersection of two sets A and B, written as A ∩ B, is the set containing all elements that are common to both A and B. If there are no common elements, the intersection is the empty set.

Properties of the Empty Set in Intersection

The intersection of the empty set with any set, including itself, is always the empty set because there are no elements to share. Thus, ∅ ∩ ∅ = ∅ is true by definition.