Finding the Complement of C

Oct 13, 2025 by recoveryshake.com 90 views

Introduction

In set theory, the union of two sets is a set containing all elements that are in either of the two sets. Given two sets A and C, we need to find the union of A and the complement of C, denoted as C'. The complement of a set is a set containing all elements that are not in the original set.

Understanding the Sets

Let's first understand the given sets:

U = {3, 7, 2, 12, 5, 13, 21, 22} is the universal set containing all elements.
A = {3, 21, 22} is a subset of U.
C = {7, 2, 12, 5, 13} is a subset of U.

Finding the Complement of C

To find the complement of C, we need to find all elements in U that are not in C.

C' = U - C = {3, 7, 2, 12, 5, 13, 21, 22} - {7, 2, 12, 5, 13} = {3, 21, 22}

Finding the Union of A and C'

Now, we need to find the union of A and C'. The union of two sets is a set containing all elements that are in either of the two sets.

A ∪ C' = {3, 21, 22} ∪ {3, 21, 22} = {3, 21, 22}

Conclusion

The union of A and C' is {3, 21, 22}. This means that the set containing all elements that are in either A or C' is {3, 21, 22}.

Final Answer

The final answer is {3, 21, 22}.

Introduction

Understanding the Sets

Let's first understand the given sets:

U = {3, 7, 2, 12, 5, 13, 21, 22} is the universal set containing all elements.
A = {3, 21, 22} is a subset of U.
C = {7, 2, 12, 5, 13} is a subset of U.

Finding the Complement of C

To find the complement of C, we need to find all elements in U that are not in C.

C' = U - C = {3, 7, 2, 12, 5, 13, 21, 22} - {7, 2, 12, 5, 13} = {3, 21, 22}

Finding the Union of A and C'

Now, we need to find the union of A and C'. The union of two sets is a set containing all elements that are in either of the two sets.

A ∪ C' = {3, 21, 22} ∪ {3, 21, 22} = {3, 21, 22}

Conclusion

The union of A and C' is {3, 21, 22}. This means that the set containing all elements that are in either A or C' is {3, 21, 22}.

Final Answer

The final answer is {3, 21, 22}.

Q&A

Q1: What is the union of two sets in set theory?

A1: The union of two sets is a set containing all elements that are in either of the two sets.

Q2: How do we find the complement of a set?

A2: To find the complement of a set, we need to find all elements in the universal set that are not in the original set.

Q3: What is the difference between the union and intersection of two sets?

A3: The union of two sets contains all elements that are in either of the two sets, while the intersection of two sets contains all elements that are in both sets.

Q4: How do we find the union of two sets?

A4: To find the union of two sets, we need to combine all elements from both sets, eliminating any duplicates.

Q5: What is the universal set in set theory?

A5: The universal set is the set containing all elements in a particular context or problem.

Q6: How do we find the complement of a set in a Venn diagram?

A6: To find the complement of a set in a Venn diagram, we need to shade the region outside the set, representing all elements that are not in the set.

Q7: What is the relationship between the union and complement of a set?

A7: The union of a set and its complement is the universal set.

Q8: How do we find the union of a set and its complement?

A8: To find the union of a set and its complement, we need to combine all elements from the set and its complement, resulting in the universal set.

Q9: What is the significance of the union of a set and its complement?

A9: The union of a set and its complement is significant because it represents the entire set of elements in a particular context or problem.

Q10: How do we apply the concept of union and complement in real-world scenarios?

A10: We can apply the concept of union and complement in real-world scenarios, such as finding the union of two groups of people, or the complement of a set of possible outcomes in a probability problem.