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

Verified step by step guidance

1. Understand the concept of the empty set, denoted as $\mathrm{\varnothing }$, which is a set with no elements.
2. Recall the definition of the intersection of two sets, $A\cap B$, which is the set of elements that are common to both sets $A$ and $B$.
3. Apply the definition of intersection to the empty set: $\mathrm{\varnothing }\cap \mathrm{\varnothing }$.
4. Since the empty set has no elements, there are no elements that can be common between the two empty sets.
5. Conclude that the intersection of two empty sets is also an empty set, $\mathrm{\varnothing }$.

Key Concepts

Set Intersection

Set intersection is a fundamental operation in set theory that identifies the common elements between two sets. The intersection of two sets A and B, denoted as A ∩ B, includes all elements that are present in both A and B. If there are no common elements, the intersection results in the empty set.

Empty Set

The empty set, denoted as ∅, is a unique set that contains no elements. It serves as the foundational building block in set theory, representing the concept of 'nothingness' in a set context. The empty set is a subset of every set and plays a crucial role in various mathematical operations.

True or False Statements

In mathematics, determining the truth value of a statement involves assessing whether the statement is accurate based on established definitions and properties. Statements can be classified as true or false, and understanding the underlying concepts is essential for making these determinations. In this case, evaluating the intersection of two empty sets requires knowledge of both set intersection and the properties of the empty set.