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′ ∩ Q

Verified step by step guidance

1. Step 1: Understand the universal set and the given subsets. The universal set is \(U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\), and the subsets are \(M = \{0, 2, 4, 6, 8\}\) and \(Q = \{0, 2, 4, 6, 8, 10, 12\}\).
2. Step 2: Find the complement of set \(M\), denoted \(M'\). The complement \(M'\) consists of all elements in \(U\) that are not in \(M\). So, \(M' = U \setminus M\).
3. Step 3: Write out the elements of \(M'\) explicitly by removing all elements of \(M\) from \(U\). This means listing all elements of \(U\) that are not in \(\{0, 2, 4, 6, 8\}\).
4. Step 4: Find the intersection of \(M'\) and \(Q\), denoted \(M' \cap Q\). This means identifying all elements that are common to both \(M'\) and \(Q\).
5. Step 5: List the elements that appear in both \(M'\) and \(Q\). These elements form the set \(M' \cap Q\). After this, check if \(M'\) and \(Q\) have any elements in common; if none, they are disjoint sets.

Key Concepts

Universal Set and Complement

The universal set U contains all elements under consideration. The complement of a set M, denoted M′, includes all elements in U that are not in M. Understanding complements helps in identifying elements outside a given subset within the universal set.

Set Intersection

The intersection of two sets, such as M′ ∩ Q, consists of all elements that are common to both sets. This operation helps find shared elements and is fundamental in analyzing relationships 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 is important for understanding mutually exclusive groups within a universal set.