Understanding the Concept of Complement of a Set

Oct 13, 2025 by recoveryshake.com 113 views

Understanding the Problem

To find the set ${$ (A \cap C)^{\prime}$}$, we first need to understand the given information and the operations involved. The universal set $U$ is defined as $\{1,2,3,4,5,6,7,8\}$ , and two subsets $A$ and $C$ are given as $A=\{1,3,4,8\}$ and $C=\{1,2,4,5,8\}$ , respectively.

Intersection of Sets

The intersection of two sets $A$ and $C$ , denoted as $A \cap C$ , is the set of elements that are common to both sets. In this case, the intersection of $A$ and $C$ is $\{1,4,8\}$ .

Complement of a Set

The complement of a set $S$ , denoted as $S^{\prime}$ , is the set of elements that are in the universal set $U$ but not in $S$ . To find the complement of $A \cap C$ , we need to find the elements that are in $U$ but not in $\{1,4,8\}$ .

Finding the Complement of $A \cap C$

The complement of $A \cap C$ is the set of elements that are in $U$ but not in $\{1,4,8\}$ . Therefore, we need to find the elements that are in $U$ but not in $\{1,4,8\}$ . The elements in $U$ but not in $\{1,4,8\}$ are $\{2,3,5,6,7\}$ .

Conclusion

Therefore, the set ${$ (A \cap C)^{\prime}$}$ is $\{2,3,5,6,7\}$ .

Final Answer

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

Understanding the Concept of Complement of a Set

The concept of complement of a set is an important concept in set theory. It is used to find the set of elements that are in the universal set but not in a given set. The complement of a set is denoted as $S^{\prime}$ , where $S$ is the given set.

Properties of Complement of a Set

The complement of a set has the following properties:

The complement of a set is a set.
The complement of a set is unique.
The complement of a set is denoted as $S^{\prime}$ , where $S$ is the given set.
The complement of a set is the set of elements that are in the universal set but not in the given set.

Example of Complement of a Set

Let $U=\{1,2,3,4,5,6,7,8\}$ and $A=\{1,3,4,8\}$ . Then, the complement of $A$ is $A^{\prime}=\{2,5,6,7\}$ .

Importance of Complement of a Set

The complement of a set is an important concept in set theory. It is used to find the set of elements that are in the universal set but not in a given set. The complement of a set is used in various applications such as probability theory, statistics, and computer science.

Conclusion

In conclusion, the set ${$ (A \cap C)^{\prime}$}$ is $\{2,3,5,6,7\}$ . The concept of complement of a set is an important concept in set theory and is used to find the set of elements that are in the universal set but not in a given set.

Final Answer

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

Understanding the Concept of Intersection of Sets

The concept of intersection of sets is an important concept in set theory. It is used to find the set of elements that are common to two or more sets. The intersection of two sets $A$ and $B$ is denoted as $A \cap B$ .

Properties of Intersection of Sets

The intersection of sets has the following properties:

The intersection of sets is a set.
The intersection of sets is unique.
The intersection of sets is denoted as $A \cap B$ , where $A$ and $B$ are the given sets.
The intersection of sets is the set of elements that are common to both sets.

Example of Intersection of Sets

Let $A=\{1,3,4,8\}$ and $C=\{1,2,4,5,8\}$ . Then, the intersection of $A$ and $C$ is $A \cap C=\{1,4,8\}$ .

Importance of Intersection of Sets

The intersection of sets is an important concept in set theory. It is used to find the set of elements that are common to two or more sets. The intersection of sets is used in various applications such as probability theory, statistics, and computer science.

Conclusion

In conclusion, the intersection of $A$ and $C$ is $\{1,4,8\}$ . The concept of intersection of sets is an important concept in set theory and is used to find the set of elements that are common to two or more sets.

Final Answer

The final answer is $\boxed{\{1,4,8\}}$ .

Understanding the Concept of Universal Set

The concept of universal set is an important concept in set theory. It is the set of all elements that are being considered in a particular problem or situation. The universal set is denoted as $U$ .

Properties of Universal Set

The universal set has the following properties:

The universal set is a set.
The universal set is unique.
The universal set is denoted as $U$ .
The universal set is the set of all elements that are being considered in a particular problem or situation.

Example of Universal Set

Let $U=\{1,2,3,4,5,6,7,8\}$ . Then, $U$ is the universal set.

Importance of Universal Set

The universal set is an important concept in set theory. It is used to define the set of all elements that are being considered in a particular problem or situation. The universal set is used in various applications such as probability theory, statistics, and computer science.

Conclusion

In conclusion, the universal set $U$ is $\{1,2,3,4,5,6,7,8\}$ . The concept of universal set is an important concept in set theory and is used to define the set of all elements that are being considered in a particular problem or situation.

Final Answer

The final answer is $\boxed{\{1,2,3,4,5,6,7,8\}}$ .

Q: What is the universal set $U$ in the given problem?

A: The universal set $U$ is defined as $\{1,2,3,4,5,6,7,8\}$ .

Q: What are the subsets $A$ and $C$ in the given problem?

A: The subsets $A$ and $C$ are given as $A=\{1,3,4,8\}$ and $C=\{1,2,4,5,8\}$ , respectively.

Q: What is the intersection of sets $A$ and $C$ ?

A: The intersection of sets $A$ and $C$ is $\{1,4,8\}$ .

Q: What is the complement of the intersection of sets $A$ and $C$ ?

A: The complement of the intersection of sets $A$ and $C$ is the set of elements that are in the universal set $U$ but not in $\{1,4,8\}$ . Therefore, the complement of the intersection of sets $A$ and $C$ is $\{2,3,5,6,7\}$ .

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{\{2,3,5,6,7\}}$ .

Q: What is the concept of complement of a set?

A: The concept of complement of a set is an important concept in set theory. It is used to find the set of elements that are in the universal set but not in a given set. The complement of a set is denoted as $S^{\prime}$ , where $S$ is the given set.

Q: What are the properties of complement of a set?

A: The complement of a set has the following properties:

The complement of a set is a set.
The complement of a set is unique.
The complement of a set is denoted as $S^{\prime}$ , where $S$ is the given set.
The complement of a set is the set of elements that are in the universal set but not in the given set.

Q: What is the importance of complement of a set?

A: The complement of a set is an important concept in set theory. It is used to find the set of elements that are in the universal set but not in a given set. The complement of a set is used in various applications such as probability theory, statistics, and computer science.

Q: What is the concept of intersection of sets?

A: The concept of intersection of sets is an important concept in set theory. It is used to find the set of elements that are common to two or more sets. The intersection of two sets $A$ and $B$ is denoted as $A \cap B$ .

Q: What are the properties of intersection of sets?

A: The intersection of sets has the following properties:

The intersection of sets is a set.
The intersection of sets is unique.
The intersection of sets is denoted as $A \cap B$ , where $A$ and $B$ are the given sets.
The intersection of sets is the set of elements that are common to both sets.

Q: What is the importance of intersection of sets?

A: The intersection of sets is an important concept in set theory. It is used to find the set of elements that are common to two or more sets. The intersection of sets is used in various applications such as probability theory, statistics, and computer science.

Q: What is the concept of universal set?

A: The concept of universal set is an important concept in set theory. It is the set of all elements that are being considered in a particular problem or situation. The universal set is denoted as $U$ .

Q: What are the properties of universal set?

A: The universal set has the following properties:

The universal set is a set.
The universal set is unique.
The universal set is denoted as $U$ .
The universal set is the set of all elements that are being considered in a particular problem or situation.

Q: What is the importance of universal set?

A: The universal set is an important concept in set theory. It is used to define the set of all elements that are being considered in a particular problem or situation. The universal set is used in various applications such as probability theory, statistics, and computer science.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{\{2,3,5,6,7\}}$ .