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′

Verified step by step guidance

1. Identify the universal set \(U\) and the subset \(Q\) given: \(U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\) and \(Q = \{0, 2, 4, 6, 8, 10, 12\}\).
2. Recall that the complement of a set \(Q\), denoted \(Q'\), consists of all elements in the universal set \(U\) that are not in \(Q\).
3. To find \(Q'\), list all elements of \(U\) that are not in \(Q\). This means you will subtract the elements of \(Q\) from \(U\).
4. Write the complement set \(Q'\) explicitly as \(Q' = U - Q = \{x \in U : x \notin Q\}\).
5. Check if \(Q'\) and \(Q\) are disjoint sets by confirming that they have no elements in common, which is always true for a set and its complement.

Key Concepts

Universal Set and Subsets

The universal set U contains all elements under consideration, while subsets like M, N, Q, and R are parts of U. Understanding the relationship between these sets helps in performing operations like complements and intersections.

Set Complement

The complement of a set Q, denoted Q′, consists of all elements in the universal set U that are not in Q. Finding Q′ involves identifying elements excluded from Q but present in U.

Disjoint Sets

Two sets are disjoint if they have no elements in common. Identifying disjoint sets requires checking for intersections that result in an empty set, which is important for understanding relationships between given sets.