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 concept of intersection: The intersection of two sets, denoted as \( A \cap B \), is the set of elements that are common to both sets.
2. Identify the empty set: The empty set, denoted as \( \emptyset \) or \( \{ \} \), is a set that contains no elements.
3. Recognize that the intersection of any set with the empty set is always the empty set, because there are no elements in the empty set to be common with any other set.
4. Apply this concept to the problem: \( \emptyset \cap Q \) is the intersection of the empty set with set \( Q \).
5. Conclude that \( \emptyset \cap Q = \emptyset \), since there are no elements in the empty set to intersect with set \( Q \).

Key Concepts

Sets and Set Notation

A set is a collection of distinct objects, considered as an object in its own right. Set notation is used to define and describe these collections, using curly braces to list elements. For example, the set M = {0, 2, 4, 6, 8} contains even numbers from 0 to 8. Understanding how to read and interpret set notation is crucial for solving problems involving sets.

Intersection of Sets

The intersection of two sets, denoted as A ∩ B, is the set of elements that are common to both A and B. For instance, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}. This concept is essential for determining shared elements between sets, which is a key part of the question.

Disjoint Sets

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