Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. N ∪ ∅

Verified step by step guidance

1. Identify the given sets: N = \{1, 3, 5, 7, 9, 11, 13\} and the empty set \(\emptyset\).
2. Recall the definition of the union of two sets: The union of two sets A and B, denoted as \(A \cup B\), is the set of elements that are in A, in B, or in both.
3. Apply the union operation to the sets N and \(\emptyset\): Since the empty set contains no elements, the union of any set with the empty set is the set itself.
4. Conclude that \(N \cup \emptyset = N\).
5. Note that the empty set is disjoint with any set that contains elements, as they have no elements in common.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In this context, understanding how to manipulate sets—such as performing unions, intersections, and identifying disjoint sets—is crucial. A union combines all elements from the involved sets, while disjoint sets have no elements in common.

Union of Sets

The union of two sets, denoted as A ∪ B, is the set containing all elements that are in A, in B, or in both. For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}. In the question, N ∪ ∅ represents the union of set N with the empty set, which will simply yield set N since the empty set contributes no elements.

Disjoint Sets

Disjoint sets are sets that have no elements in common. For instance, if set A = {1, 2} and set B = {3, 4}, then A and B are disjoint. In the provided sets, identifying disjoint sets involves checking for any overlap in their elements, which is essential for understanding their relationships and performing operations like unions or intersections.