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. N ∪ ∅

Verified step by step guidance

1. Understand the problem: You are given several sets and asked to find the union of set N and the empty set ∅, and also to identify any disjoint sets among the given ones.
2. Recall the definition of union of two sets: The union of sets A and B, denoted by \(A \cup B\), is the set containing all elements that are in A, or in B, or in both.
3. Recall the property of the empty set: The empty set \(\emptyset\) has no elements, so the union of any set with \(\emptyset\) is just the original set itself.
4. Apply the union operation: Since \(N = \{1, 3, 5, 7, 9, 11, 13\}\) and \(\emptyset\) has no elements, \(N \cup \emptyset = N\).
5. To identify disjoint sets, check pairs of sets to see if they have no elements in common. Two sets are disjoint if their intersection is the empty set.

Key Concepts

Set Union

The union of two sets combines all elements from both sets without duplication. For example, the union of set N and the empty set ∅ includes all elements of N, since ∅ has no elements. Union is denoted by the symbol ∪.

Empty Set (∅)

The empty set is a set containing no elements. It acts as the identity element for union, meaning any set unioned with ∅ remains unchanged. Understanding the empty set is crucial for operations involving sets.

Disjoint Sets

Disjoint sets are sets that have no elements in common. Identifying disjoint sets involves checking if their intersection is empty. This concept helps in understanding relationships between sets and their elements.