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 ∩ R′

Verified step by step guidance

1. Step 1: Understand the problem and the given sets. The universal set is \(U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\), and the sets are \(M\), \(N\), \(Q\), and \(R\) as defined. We need to find \(Q \cap R'\) where \(R'\) is the complement of \(R\) relative to \(U\).
2. Step 2: Find the complement of \(R\), denoted \(R'\). The complement \(R'\) consists of all elements in \(U\) that are not in \(R\). So, \(R' = U \setminus R\).
3. Step 3: Write out the elements of \(R'\) explicitly by removing all elements of \(R\) from \(U\). Since \(R = \{0, 1, 2, 3, 4\}\), remove these from \(U\) to get \(R'\).
4. Step 4: Find the intersection \(Q \cap R'\). This means identifying all elements that are both in \(Q\) and in \(R'\). Recall that \(Q = \{0, 2, 4, 6, 8, 10, 12\}\).
5. Step 5: After finding \(Q \cap R'\), check if the resulting set is disjoint with any of the given sets by verifying if they share any common elements.

Key Concepts

Set Complement

The complement of a set R, denoted R′, consists of all elements in the universal set U that are not in R. It helps identify elements outside a particular subset, which is essential for operations like intersections involving complements.

Set Intersection

The intersection of two sets, such as Q ∩ R′, includes all elements common to both sets. Understanding intersection allows you to find shared elements between a set and the complement of another.

Disjoint Sets

Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set. Identifying disjoint sets helps in understanding relationships and exclusivity between different subsets.