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

Verified step by step guidance

1. Recall the definition of the union of two sets: for any sets A and B, the union A \cup B is the set containing all elements that are in A, or in B, or in both.
2. Identify the sets involved: here both sets are the empty set, denoted by \emptyset, which contains no elements.
3. Apply the union operation to the empty sets: \emptyset \cup \emptyset means the set of all elements in \emptyset or in \emptyset.
4. Since \emptyset has no elements, there are no elements to combine, so the union \emptyset \cup \emptyset is also an empty set.
5. Conclude that the statement \emptyset \cup \emptyset = \emptyset is true because the union of two empty sets is still an empty set.

Key Concepts

Empty Set (∅)

The empty set, denoted by ∅, is the set containing no elements. It is unique and serves as the foundation for set theory. Understanding that ∅ has no members is crucial for evaluating operations involving it.

Union of Sets

The union of two sets A and B, written as A ∪ B, is the set containing all elements that are in A, in B, or in both. If both sets are empty, their union will also be empty since there are no elements to combine.

Set Equality

Two sets are equal if and only if they contain exactly the same elements. To determine if ∅ ∪ ∅ = ∅, we check if both sides have the same elements, which they do since both are empty sets.