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

Verified step by step guidance

1. Recall the definition of the intersection of two sets: the intersection \(A \cap B\) is the set of all elements that are common to both sets \(A\) and \(B\).
2. Identify the two sets in the problem: \(A = \{3, 5, 9, 10\}\) and \(B = \emptyset\) (the empty set).
3. Since \(B\) is the empty set, it contains no elements, so there are no elements that can be common to both \(A\) and \(B\).
4. Therefore, the intersection \(A \cap B\) must be the empty set \(\emptyset\), because there are no shared elements.
5. Compare this result to the statement given: \(\{3, 5, 9, 10\} \cap \emptyset = \{3, 5, 9, 10\}\). Since the intersection is actually \(\emptyset\), the statement is false.

Key Concepts

Set Intersection

The intersection of two sets includes all elements that are common to both sets. For example, the intersection of {1, 2, 3} and {2, 3, 4} is {2, 3}. Understanding this helps determine which elements belong to both sets simultaneously.

Empty Set (∅)

The empty set is a set with no elements. It is unique and is a subset of every set. Since it contains nothing, any intersection with the empty set results in the empty set itself.

Properties of Set Operations

Set operations follow specific properties, such as the intersection of any set with the empty set being the empty set. Recognizing these properties allows for quick evaluation of set expressions without listing elements.