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

Verified step by step guidance

1. Understand the symbols: The symbol $\mathrm{\varnothing }$ represents the empty set, which is a set with no elements.
2. Recall the definition of union: The union of two sets $A$ and $B$, denoted $A\cup B$, is the set of elements that are in $A$, in $B$, or in both.
3. Apply the union operation to the empty sets: Since both sets are empty, there are no elements in either set to combine.
4. Conclude the result of the union: The union of two empty sets is still an empty set because there are no elements to include in the union.
5. Determine the truth value: Since $\mathrm{\varnothing }\cup \mathrm{\varnothing }=\mathrm{\varnothing }$ is a true statement, the given statement is true.

Key Concepts

Set Union

Set union is an operation that combines all unique elements from two or more sets. The union of sets A and B, denoted as A ∪ B, includes every element that is in A, in B, or in both. Understanding this concept is crucial for determining the outcome of union operations, especially when dealing with empty sets.

Empty Set

The empty set, denoted as ∅, is a set that contains no elements. It is a fundamental concept in set theory, serving as the identity element for the union operation. Recognizing the properties of the empty set is essential for evaluating statements involving unions and intersections.

Identity Element in Set Operations

In set theory, the identity element for union is the empty set. This means that when you take the union of any set with the empty set, the result is the original set. This property is vital for understanding how operations on sets behave, particularly in confirming the truth of statements involving unions of empty sets.