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. M ∪ N

Verified step by step guidance

1. Identify the elements in set M: \( M = \{0, 2, 4, 6, 8\} \).
2. Identify the elements in set N: \( N = \{1, 3, 5, 7, 9, 11, 13\} \).
3. The union of two sets, \( M \cup N \), includes all elements that are in either set M or set N or in both.
4. List all unique elements from both sets M and N.
5. Check if there are any common elements between sets M and N to determine if they are disjoint. If there are no common elements, the sets are disjoint.

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 the sets M and N provided in the question, the union will include all even and odd numbers from the specified ranges. This concept is fundamental for combining sets to analyze their collective elements.

Disjoint Sets

Disjoint sets are sets that have no elements in common, meaning their intersection is the empty set. Identifying disjoint sets is important in various applications, including probability and statistics. In the given sets, M and N are disjoint because they contain only even and odd numbers, respectively, with no overlap.