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 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 in the problem: A = {3, 5, 9, 10} and B = \(\emptyset\) (the empty set, which contains no elements).
3. Since the empty set has no elements, adding it to any set A through union does not add any new elements to A.
4. Therefore, the union {3, 5, 9, 10} \(\cup\) \(\emptyset\) will contain exactly the elements of {3, 5, 9, 10}, with no additions or removals.
5. Conclude that the statement {3, 5, 9, 10} \(\cup\) \(\emptyset\) = {3, 5, 9, 10} is true based on the properties of the empty set and union operation.

Key Concepts

Set Union

The union of two sets combines all unique elements from both sets into one set. For example, the union of {1, 2} and {2, 3} is {1, 2, 3}. Understanding union helps determine the result when combining sets.

Empty Set (∅)

The empty set is a set with no elements. It is unique and often used as the identity element in set operations. When unioned with any set, the result is the original set because no new elements are added.

Set Equality

Two sets are equal if they contain exactly the same elements, regardless of order or repetition. This concept is essential to verify if the union operation results in the given set.