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. Q ∩ (M ∪ N)

Verified step by step guidance

1. First, understand the problem: You are asked to find the intersection of set Q with the union of sets M and N, which is written as \(Q \cap (M \cup N)\).
2. Step 1: Find the union of sets M and N. The union \(M \cup N\) includes all elements that are in M, or in N, or in both. Write down all unique elements from M and N combined.
3. Step 2: Once you have \(M \cup N\), find the intersection with Q. The intersection \(Q \cap (M \cup N)\) includes only those elements that are in both Q and \(M \cup N\).
4. Step 3: List the elements of Q and compare them with the elements of \(M \cup N\) to identify the common elements.
5. Step 4: After finding the intersection, check if any of the sets involved are disjoint. Two sets are disjoint if their intersection is the empty set (no common elements).

Key Concepts

Set Union ( ∪ )

The union of two sets combines all elements from both sets without duplication. For example, if M = {0, 2, 4} and N = {1, 3, 5}, then M ∪ N = {0, 1, 2, 3, 4, 5}. This operation helps in gathering all unique elements from the involved sets.

Set Intersection ( ∩ )

The intersection of two sets includes only the elements that appear in both sets. For instance, if Q = {0, 2, 4} and M ∪ N = {0, 1, 2, 3, 4, 5}, then Q ∩ (M ∪ N) = {0, 2, 4}. This concept is used to find common elements between sets.

Disjoint Sets

Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set (∅). Identifying disjoint sets helps understand relationships between sets, such as whether they share any elements or are completely separate.