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 ∉ R}

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 \(R\). This is the set difference \(U \setminus R\).
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 \(B\). Symbolically, \(A \setminus B = \{ x \mid x \in A \text{ and } x \notin B \}\).
3. Identify the elements of \(U\) and \(R\): \(U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}\) and \(R = \{0, 1, 2, 3, 4\}\).
4. Find the elements in \(U\) that are not in \(R\): Go through each element of \(U\) and exclude those that appear in \(R\). This will give the set \(\{ x \mid x \in U, x \notin R \}\).
5. After finding the set difference, check if this resulting set is disjoint with any of the other given sets \(M\), \(N\), \(Q\), or \(R\). Two sets are disjoint if they have no elements in common.

Key Concepts

Set Notation and Membership

Set notation uses symbols to describe collections of elements. The symbol '∈' means 'is an element of,' and '∉' means 'is not an element of.' Understanding these symbols helps interpret expressions like {x | x ∈ U, x ∉ R}, which describes all elements x in set U that are not in set R.

Set Difference

The set difference between two sets A and B, denoted A \ B, consists of elements in A that are not in B. In this problem, {x | x ∈ U, x ∉ R} represents the difference U \ R, which means removing all elements of R from U to find the remaining elements.

Disjoint Sets

Two sets are disjoint if they have no elements in common. Identifying disjoint sets involves checking for intersections that are empty. This concept is important here to determine which given sets share no common elements.