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. {x | x ∈ M and x ∈ Q}

Verified step by step guidance

1. Understand the problem: We are given several sets and asked to find the set of elements \( x \) such that \( x \in M \) and \( x \in Q \). This means we need to find the intersection of sets \( M \) and \( Q \).
2. Recall the definition of intersection: The intersection of two sets \( A \) and \( B \), denoted \( A \cap B \), is the set of all elements that are in both \( A \) and \( B \). So, \( M \cap Q = \{ x \mid x \in M \text{ and } x \in Q \} \).
3. List the elements of each set explicitly: \( M = \{0, 2, 4, 6, 8\} \) and \( Q = \{0, 2, 4, 6, 8, 10, 12\} \).
4. Compare the elements of \( M \) and \( Q \) to find common elements. Identify which elements appear in both sets.
5. Write the intersection set \( M \cap Q \) as the set of all common elements found in the previous step. Then, check if \( M \) and \( Q \) are disjoint by verifying if their intersection is empty or not.

Key Concepts

Set Intersection

The intersection of two sets includes all elements that are common to both sets. It is denoted by the symbol ∩. For example, if M = {0, 2, 4} and Q = {2, 4, 6}, then M ∩ Q = {2, 4}. Understanding intersection helps identify shared elements between sets.

Disjoint Sets

Disjoint sets are sets that have no elements in common, meaning their intersection is the empty set (∅). For instance, if A = {1, 3} and B = {2, 4}, then A and B are disjoint because A ∩ B = ∅. Recognizing disjoint sets is important for understanding relationships between sets.

Set Notation and Membership

Set notation uses symbols like ∈ to indicate membership, meaning an element belongs to a set. For example, x ∈ M means x is an element of set M. Proper understanding of notation is essential to interpret and solve problems involving sets accurately.