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

Verified step by step guidance

1. Understand the problem: You are given several sets and asked to find the intersection of the empty set ∅ and the set Q, and to identify any disjoint sets. The empty set ∅ contains no elements.
2. Recall the definition of intersection: The intersection of two sets A and B, denoted by \(A \cap B\), is the set of all elements that are common to both A and B.
3. Apply the intersection operation to the empty set and Q: Since the empty set has no elements, there are no elements that can be common with any other set. Therefore, \(\emptyset \cap Q = \emptyset\).
4. Interpret the result: The intersection of the empty set with any set is always the empty set, which means they have no elements in common.
5. Identify disjoint sets: Two sets are disjoint if their intersection is the empty set. Since \(\emptyset\) has no elements, it is disjoint with every set, including Q.

Key Concepts

Set Intersection

The intersection of two sets consists of all elements that are common to both sets. It is denoted by the symbol ∩. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}. Understanding intersection helps identify shared elements between sets.

Empty Set (∅)

The empty set, denoted by ∅, is a set that contains no elements. It is unique and is a subset of every set. When intersecting any set with ∅, the result is always ∅ because there are no elements to share.

Disjoint Sets

Two sets are disjoint if they have no elements in common, meaning their intersection is the empty set. Identifying disjoint sets is important for understanding relationships between sets and for solving problems involving unions and intersections.