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. (U ∩ ∅′) ∪ R

Verified step by step guidance

1. Step 1: Understand the universal set U and the empty set ∅. The universal set U contains all elements under consideration, and the empty set ∅ has no elements. The complement of the empty set, denoted as ∅′, is the set of all elements in U that are not in ∅. Since ∅ has no elements, ∅′ = U.
2. Step 2: Calculate the intersection of U and ∅′, which is U ∩ ∅′. Since ∅′ = U, the intersection U ∩ ∅′ is simply U itself.
3. Step 3: Recall the set R = {0, 1, 2, 3, 4}. The problem asks for (U ∩ ∅′) ∪ R, which simplifies to U ∪ R because U ∩ ∅′ = U.
4. Step 4: Since U contains all elements from 0 to 13, and R is a subset of U, the union U ∪ R is just U. So the resulting set is U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}.
5. Step 5: To identify any disjoint sets among the given sets, recall that two sets are disjoint if they have no elements in common. Check pairs such as M and N, M and Q, N and R, etc., by comparing their elements to see if their intersection is empty.

Key Concepts

Universal Set and Subsets

The universal set U contains all elements under consideration, while subsets like M, N, Q, and R are parts of U. Understanding the relationship between the universal set and its subsets helps in performing set operations and interpreting results within the defined context.

Set Complement

The complement of a set, denoted by ∅′, includes all elements in the universal set that are not in the given set. Since ∅ is the empty set, its complement is the entire universal set U. Recognizing complements is essential for correctly evaluating expressions involving complements.

Set Operations: Intersection and Union

Intersection (∩) finds common elements between sets, while union (∪) combines all elements from the involved sets without duplication. Applying these operations step-by-step allows for simplifying complex set expressions and identifying relationships such as disjointness.