Let \( M=\{10,12,14,16,18\}, Q=\{10,12,14,16,18,20,22\} \), and \( R=\{10,11,12,13,14\} \). Find ( \( Q \cap M) \cup R \). Identify any disjoint sets.

To solve the problem, we first need to find the intersection of the sets \( Q \) and \( M \). The intersection is given by: \[ Q \cap M = \{10, 12, 14, 16, 18\} \cap \{10, 12, 14, 16, 18, 20, 22\} = \{10, 12, 14, 16, 18\} \] Next, we take the union of this intersection with set \( R \): \[ (Q \cap M) \cup R = \{10, 12, 14, 16, 18\} \cup \{10, 11, 12, 13, 14\} \] To calculate this union, we combine the elements from both sets without repetition: \[ (Q \cap M) \cup R = \{10, 11, 12, 13, 14, 16, 18\} \] Now, we identify any disjoint sets. A disjoint set is one that shares no elements with another set. Observing our sets: 1. **Set \( M = \{10, 12, 14, 16, 18\} \)** and **Set \( R = \{10, 11, 12, 13, 14\} \)** share elements (10, 12, and 14). 2. **Set \( Q \)** shares all elements with \( M \), thus not disjoint. 3. However, **Set \( R \)** contains the element 11 which is not in \( M\) and neither in \( Q \). This shows that \( R \) has elements (11 and 13) not present in \( Q \cap M \), confirming that **R is disjoint with \{16, 18\}**. So the final findings are: \[ (Q \cap M) \cup R = \{10, 11, 12, 13, 14, 16, 18\} \] And the disjoint sets from this analysis can be mentioned as: - Set \( R \) (since it contains elements like 11 and 13 that are not in the intersection of \( Q \) and \( M \)). This is a fun and engaging way to visualize the relationship between these sets!