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 ∈ U, x ∉ M}

Verified step by step guidance

1. Understand the problem: We are given a universal set \(U\) and several subsets \(M\), \(N\), \(Q\), and \(R\). The task is to find the set of all elements \(x\) such that \(x\) is in \(U\) but not in \(M\). This is the set difference \(U \setminus M\).
2. Recall the definition of set difference: For two sets \(A\) and \(B\), the difference \(A \setminus B\) is the set of all elements that are in \(A\) but not in \(B\). So here, \(U \setminus M = \{ x \mid x \in U \text{ and } x \notin M \}\).
3. List the elements of \(U\) and \(M\) explicitly: \(U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\) and \(M = \{0, 2, 4, 6, 8\}\).
4. Remove all elements of \(M\) from \(U\): Go through each element of \(U\) and exclude those that appear in \(M\). The remaining elements form the set \(U \setminus M\).
5. After finding \(U \setminus M\), analyze the sets to identify any disjoint sets. Two sets are disjoint if they have no elements in common. Compare \(U \setminus M\) with the other given sets \(N\), \(Q\), and \(R\) to check for disjointness.

Key Concepts

Set Notation and Set Builder Notation

Set notation is a way to describe collections of elements, while set builder notation specifies a set by stating the properties its members must satisfy. For example, {x | x ∈ U, x ∉ M} means the set of all elements x that belong to U but not to M. Understanding this helps interpret and form sets based on given conditions.

Set Operations: Complement and Difference

The expression {x | x ∈ U, x ∉ M} represents the complement of M relative to U, or equivalently, the difference U \ M. This operation includes all elements in U that are not in M. Mastery of set difference and complement is essential for manipulating and finding specific subsets.

Disjoint Sets

Two sets are disjoint if they have no elements in common. Identifying disjoint sets involves checking for intersections that are empty. Recognizing disjointness is important for understanding relationships between sets and for solving problems involving unions and intersections.