Finding the Intersection of A and C

Oct 13, 2025 by magicalceylontours.com 70 views

Understanding the Problem

In set theory, the intersection of two sets is the set of elements that are common to both sets. The complement of a set is the set of elements that are not in the original set. Given two sets, A and C, we are asked to find the complement of the intersection of A and C, denoted as ${$ (A \cap C)^{\prime}$}$.

The Sets A and C

To solve this problem, we need to understand the elements of sets A and C. The universal set U is given as {1,2,3,4,5,6,7,8}. Set A is defined as {1,3,4,8} and set C is defined as {1,2,4,5,8}.

Finding the Intersection of A and C

The intersection of A and C, denoted as ${$ A \cap C$}$, is the set of elements that are common to both A and C. To find the intersection, we need to identify the elements that are present in both sets.

A = {1,3,4,8}
C = {1,2,4,5,8}

The elements that are common to both sets are 1, 4, and 8. Therefore, the intersection of A and C is {1,4,8}.

Finding the Complement of the Intersection

The complement of a set is the set of elements that are not in the original set. To find the complement of the intersection of A and C, we need to identify the elements that are not in {1,4,8}.

U = {1,2,3,4,5,6,7,8}
A ∩ C = {1,4,8}

The elements that are not in {1,4,8} are 2, 3, 5, 6, and 7. Therefore, the complement of the intersection of A and C is {2,3,5,6,7}.

Conclusion

In conclusion, the set ${$ (A \cap C)^{\prime}$}$ is the complement of the intersection of A and C, which is {2,3,5,6,7}.

Final Answer

The final answer is: $\boxed{\{2,3,5,6,7\}}$

Discussion

This problem requires a basic understanding of set theory, including the concepts of intersection and complement. The intersection of two sets is the set of elements that are common to both sets, while the complement of a set is the set of elements that are not in the original set. By applying these concepts, we can find the complement of the intersection of A and C.

Example Problems

Find the set ${$ (A \cup C)^{\prime}$}$ given that A = {1,3,4,8} and C = {1,2,4,5,8}.
Find the set ${$ (A \cap B)^{\prime}$}$ given that A = {1,3,4,8} and B = {2,3,4,5}.

Solutions

Find the set ${$ (A \cup C)^{\prime}$}$ given that A = {1,3,4,8} and C = {1,2,4,5,8}. The union of A and C is {1,2,3,4,5,8}. The complement of the union of A and C is {6,7}.
Find the set ${$ (A \cap B)^{\prime}$}$ given that A = {1,3,4,8} and B = {2,3,4,5}. The intersection of A and B is {3,4}. The complement of the intersection of A and B is {1,2,5,8}.

Understanding Set Theory and Complements

Set theory is a branch of mathematics that deals with the study of sets, which are collections of unique objects. In this article, we will explore some common questions and answers related to set theory and complements.

Q: What is the intersection of two sets?

A: The intersection of two sets is the set of elements that are common to both sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the intersection of A and C is {1,4,8}.

Q: What is the complement of a set?

A: The complement of a set is the set of elements that are not in the original set. For example, if we have a set A = {1,3,4,8} and the universal set U = {1,2,3,4,5,6,7,8}, the complement of A is {2,5,6,7}.

Q: How do you find the complement of the intersection of two sets?

A: To find the complement of the intersection of two sets, you need to identify the elements that are not in the intersection of the two sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the intersection of A and C is {1,4,8}. The complement of the intersection of A and C is the set of elements that are not in {1,4,8}, which is {2,3,5,6,7}.

Q: What is the difference between the union and intersection of two sets?

A: The union of two sets is the set of elements that are in either set, while the intersection of two sets is the set of elements that are common to both sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the union of A and C is {1,2,3,4,5,8} and the intersection of A and C is {1,4,8}.

Q: How do you find the complement of the union of two sets?

A: To find the complement of the union of two sets, you need to identify the elements that are not in the union of the two sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the union of A and C is {1,2,3,4,5,8}. The complement of the union of A and C is the set of elements that are not in {1,2,3,4,5,8}, which is {6,7}.

Q: What is the relationship between the intersection and complement of a set?

A: The intersection and complement of a set are related in that the intersection of a set and its complement is the empty set. For example, if we have a set A = {1,3,4,8} and the universal set U = {1,2,3,4,5,6,7,8}, the intersection of A and its complement is the empty set {}.

Q: How do you find the intersection of a set and its complement?

A: To find the intersection of a set and its complement, you need to identify the elements that are common to both the set and its complement. However, since the intersection of a set and its complement is the empty set, there are no elements that are common to both the set and its complement.

Q: What is the relationship between the union and complement of a set?

A: The union and complement of a set are related in that the union of a set and its complement is the universal set. For example, if we have a set A = {1,3,4,8} and the universal set U = {1,2,3,4,5,6,7,8}, the union of A and its complement is the universal set U = {1,2,3,4,5,6,7,8}.

Q: How do you find the union of a set and its complement?

A: To find the union of a set and its complement, you need to identify the elements that are in either the set or its complement. Since the union of a set and its complement is the universal set, all elements in the universal set are in the union of the set and its complement.

Q: What is the relationship between the intersection and union of two sets?

A: The intersection and union of two sets are related in that the intersection of two sets is a subset of the union of the two sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the intersection of A and C is {1,4,8} and the union of A and C is {1,2,3,4,5,8}. The intersection of A and C is a subset of the union of A and C.

Q: How do you find the intersection of two sets and the union of the two sets?

A: To find the intersection of two sets, you need to identify the elements that are common to both sets. To find the union of two sets, you need to identify the elements that are in either set.

Q: What is the relationship between the complement and union of two sets?

A: The complement and union of two sets are related in that the complement of the union of two sets is the intersection of the complements of the two sets. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the union of A and C is {1,2,3,4,5,8} and the complement of the union of A and C is {6,7}. The complement of the union of A and C is the intersection of the complements of A and C.

Q: How do you find the complement of the union of two sets and the intersection of the complements of the two sets?

A: To find the complement of the union of two sets, you need to identify the elements that are not in the union of the two sets. To find the intersection of the complements of the two sets, you need to identify the elements that are common to both the complements of the two sets.

Q: What is the relationship between the intersection and complement of the union of two sets?

A: The intersection and complement of the union of two sets are related in that the intersection of the union of two sets and its complement is the empty set. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the union of A and C is {1,2,3,4,5,8} and the complement of the union of A and C is {6,7}. The intersection of the union of A and C and its complement is the empty set {}.

Q: How do you find the intersection of the union of two sets and its complement?

A: To find the intersection of the union of two sets and its complement, you need to identify the elements that are common to both the union of the two sets and its complement. However, since the intersection of the union of two sets and its complement is the empty set, there are no elements that are common to both the union of the two sets and its complement.

Q: What is the relationship between the union and complement of the intersection of two sets?

A: The union and complement of the intersection of two sets are related in that the union of the intersection of two sets and its complement is the universal set. For example, if we have two sets A = {1,3,4,8} and C = {1,2,4,5,8}, the intersection of A and C is {1,4,8} and the complement of the intersection of A and C is {2,3,5,6,7}. The union of the intersection of A and C and its complement is the universal set {1,2,3,4,5,6,7,8}.

Q: How do you find the union of the intersection of two sets and its complement?

A: To find the union of the intersection of two sets and its complement, you need to identify the elements that are in either the intersection of the two sets or its complement. Since the union of the intersection of two sets and its complement is the universal set, all elements in the universal set are in the union of the intersection of the two sets and its complement.

Q: What is the relationship between the intersection and union of the complement of the intersection of two sets?

A: The intersection and union of the complement of the intersection of two sets are related in that the intersection of the complement of the intersection of two sets and its complement is the empty set. For example, if we have two sets A = {1,3,4,