Determine whether each statement is true or false. {3, 5, 9, 10} ∩ ∅ = {3, 5, 9, 10}

Verified step by step guidance

1. Identify the sets involved in the problem. Here, we have two sets: {3, 5, 9, 10} and the empty set ∅.
2. Understand the operation being performed. The symbol '∩' represents the intersection of two sets, which includes only the elements that are common to both sets.
3. Analyze the empty set ∅. The empty set is a set with no elements.
4. Determine the intersection of {3, 5, 9, 10} and ∅. Since there are no elements in the empty set, there are no common elements between {3, 5, 9, 10} and ∅.
5. Conclude that the intersection of {3, 5, 9, 10} and ∅ is itself an empty set, represented as ∅.

Key Concepts

Set Intersection

Set intersection is a fundamental operation in set theory that identifies the common elements between two sets. The intersection of sets A and B, denoted as A ∩ B, includes only those elements that are present in both sets. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.

Empty Set

The empty set, denoted as ∅, is a unique set that contains no elements. It serves as the identity element for set union and the annihilator for set intersection. When any set is intersected with the empty set, the result is always the empty set, as there are no common elements to include.

Truth Values in Set Theory

In set theory, statements about sets can be evaluated as true or false based on their definitions and properties. For instance, the statement 'A ∩ B = C' is true if the intersection of sets A and B equals set C. Understanding how to evaluate these statements is crucial for determining the validity of claims involving sets.