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) ∪ R

Verified step by step guidance

1. Identify the sets involved in the expression: M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, and R = {0, 1, 2, 3, 4}.
2. Find the intersection of sets M and N, denoted as \(M \cap N\). The intersection contains elements that are in both M and N.
3. Since M contains even numbers and N contains odd numbers, determine if there are any common elements between M and N.
4. Take the union of the result from \(M \cap N\) with set R, denoted as \((M \cap N) \cup R\). The union contains all elements that are in either set.
5. Check if any of the sets M, N, or R are disjoint by verifying if their intersections are empty sets.

Key Concepts

Set Intersection (∩)

The intersection of two sets includes all elements that are common to both sets. For example, if M = {0, 2, 4} and N = {2, 3, 4}, then M ∩ N = {2, 4}. This operation helps identify shared elements between sets.

Set Union (∪)

The union of two sets combines all elements from both sets without duplication. For instance, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}. Union is used to merge sets into a single set containing all unique elements.

Disjoint Sets

Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set. For example, sets {1, 3} and {2, 4} are disjoint. Identifying disjoint sets helps understand relationships and overlaps between groups.