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. Step 1: Understand the problem. We are given a universal set U and several subsets, including Q. We need to find the complement of Q, denoted as Q′.
2. Step 2: Recall the definition of a complement. The complement of a set Q, denoted as Q′, consists of all elements in the universal set U that are not in Q.
3. Step 3: List the elements of the universal set U: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}.
4. Step 4: List the elements of the set Q: {0, 2, 4, 6, 8, 10, 12}.
5. Step 5: Identify the elements in U that are not in Q to find Q′. These elements will be the complement of Q.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In this context, understanding how to manipulate and analyze sets is crucial, including operations like union, intersection, and complement. Familiarity with these operations allows for effective problem-solving when working with multiple sets.

Complement of a Set

The complement of a set A, denoted as A′, consists of all elements in the universal set U that are not in A. For example, if U is the universal set and Q is a subset of U, then Q′ includes all elements in U that are not in Q. This concept is essential for determining the elements outside a given set.

Disjoint Sets

Disjoint sets are sets that have no elements in common, meaning their intersection is empty. Identifying disjoint sets is important in set theory as it helps in understanding relationships between different sets. For instance, if M and N are disjoint, then M ∩ N = ∅, which simplifies many operations and analyses involving these sets.