Verifying Set Theory Identities With Factors Of 12, 15, And 18
In the realm of mathematics, set theory stands as a foundational pillar, providing the language and tools to describe and manipulate collections of objects. Understanding set operations and their properties is crucial for various branches of mathematics and computer science. This article delves into the verification of several fundamental set theory identities, offering a detailed exploration of each identity with the sets A, B, and C defined as follows:
- A = x
- B = x
- C = x
We will meticulously examine four key identities, providing step-by-step explanations and justifications for each. These identities showcase the interplay between set operations such as union, intersection, difference, and complement, offering valuable insights into the structure and behavior of sets.
Defining the Sets
Before diving into the identities, let's explicitly define the elements of each set based on the given conditions. This will provide a concrete foundation for our subsequent verifications. Understanding the elements within each set is critical for accurately performing set operations and validating the identities.
Set A: Factors of 12
Set A comprises all positive integers that divide 12 without leaving a remainder. To identify these factors, we can systematically check each integer from 1 to 12. The factors of 12 are 1, 2, 3, 4, 6, and 12. Thus, we can represent set A as:
A = {1, 2, 3, 4, 6, 12}
These elements form the building blocks for understanding how set A interacts with other sets in the identities we will explore. Each number in this set plays a crucial role in determining the outcomes of set operations involving A.
Set B: Factors of 15
Set B consists of all positive integers that divide 15 without leaving a remainder. Similar to set A, we can find these factors by checking integers from 1 to 15. The factors of 15 are 1, 3, 5, and 15. Therefore, set B can be represented as:
B = {1, 3, 5, 15}
This set, with its unique combination of elements, will interact with sets A and C in various ways, leading to different results based on the set operations applied. The interplay of these elements will be a key focus in our verification process.
Set C: Factors of 18
Set C includes all positive integers that divide 18 without leaving a remainder. We identify these factors by checking integers from 1 to 18. The factors of 18 are 1, 2, 3, 6, 9, and 18. Hence, set C is represented as:
C = {1, 2, 3, 6, 9, 18}
With a different set of factors compared to sets A and B, set C will contribute its unique elements to the set operations, influencing the final results and helping us verify the identities in question. Understanding the composition of this set is essential for accurate analysis.
Identity (i): A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
The first identity we aim to verify is the distributive law of intersection over union, expressed as:
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
This identity suggests that the intersection of set A with the union of sets B and C is equivalent to the union of the intersections of A with B and A with C. To verify this, we will compute both sides of the equation separately and demonstrate their equality. This involves several steps, including finding the union of B and C, the intersections of A with B and A with C, and finally, comparing the results.
Left-Hand Side: A ∩ (B ∪ C)
First, we need to find the union of sets B and C, denoted as B ∪ C. The union of two sets is a new set that contains all the elements present in either set. Given B = {1, 3, 5, 15} and C = {1, 2, 3, 6, 9, 18}, their union is:
B ∪ C = {1, 2, 3, 5, 6, 9, 15, 18}
This set includes all unique elements from both B and C. Next, we find the intersection of set A with this union. The intersection of two sets contains only the elements that are common to both sets. Given A = {1, 2, 3, 4, 6, 12} and B ∪ C = {1, 2, 3, 5, 6, 9, 15, 18}, their intersection is:
A ∩ (B ∪ C) = {1, 2, 3, 6}
This set represents the elements that are present in both A and the union of B and C. This result forms the left-hand side of our identity, which we will compare with the right-hand side.
Right-Hand Side: (A ∩ B) ∪ (A ∩ C)
To compute the right-hand side, we first find the intersection of A and B, denoted as A ∩ B. Given A = {1, 2, 3, 4, 6, 12} and B = {1, 3, 5, 15}, their intersection is:
A ∩ B = {1, 3}
This set contains the elements common to both A and B. Next, we find the intersection of A and C, denoted as A ∩ C. Given A = {1, 2, 3, 4, 6, 12} and C = {1, 2, 3, 6, 9, 18}, their intersection is:
A ∩ C = {1, 2, 3, 6}
This set includes the elements common to both A and C. Finally, we find the union of these two intersections. The union of A ∩ B and A ∩ C is:
(A ∩ B) ∪ (A ∩ C) = {1, 3} ∪ {1, 2, 3, 6} = {1, 2, 3, 6}
This set contains all unique elements from both A ∩ B and A ∩ C. This result represents the right-hand side of our identity.
Verification
Comparing the left-hand side and the right-hand side, we have:
- A ∩ (B ∪ C) = {1, 2, 3, 6}
- (A ∩ B) ∪ (A ∩ C) = {1, 2, 3, 6}
Since both sides are equal, we have successfully verified the identity:
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
This verification demonstrates the distributive property of intersection over union for the given sets A, B, and C. The detailed steps and calculations provide a clear understanding of how the identity holds true.
Identity (ii): A - (B ∪ C) = (A - B) ∩ (A - C)
The second identity we will explore is related to the set difference operation and its interaction with union and intersection. The identity states:
A - (B ∪ C) = (A - B) ∩ (A - C)
This identity expresses that the set difference between A and the union of B and C is equal to the intersection of the set differences between A and B, and A and C. To verify this, we will compute both sides of the equation independently and show their equality. This involves finding the union of B and C, the set differences A - B and A - C, and the final intersection and set difference.
Left-Hand Side: A - (B ∪ C)
We have already calculated the union of B and C in the previous verification:
B ∪ C = {1, 2, 3, 5, 6, 9, 15, 18}
Now, we need to find the set difference between A and B ∪ C, denoted as A - (B ∪ C). The set difference A - (B ∪ C) contains elements that are in A but not in B ∪ C. Given A = {1, 2, 3, 4, 6, 12} and B ∪ C = {1, 2, 3, 5, 6, 9, 15, 18}, the set difference is:
A - (B ∪ C) = {4, 12}
This set includes elements from A that are not present in B ∪ C. This result constitutes the left-hand side of the identity.
Right-Hand Side: (A - B) ∩ (A - C)
To compute the right-hand side, we first find the set difference between A and B, denoted as A - B. This set contains elements that are in A but not in B. Given A = {1, 2, 3, 4, 6, 12} and B = {1, 3, 5, 15}, the set difference is:
A - B = {2, 4, 6, 12}
Next, we find the set difference between A and C, denoted as A - C. This set contains elements that are in A but not in C. Given A = {1, 2, 3, 4, 6, 12} and C = {1, 2, 3, 6, 9, 18}, the set difference is:
A - C = {4, 12}
Finally, we find the intersection of these two set differences. The intersection of A - B and A - C is:
(A - B) ∩ (A - C) = {2, 4, 6, 12} ∩ {4, 12} = {4, 12}
This set contains elements that are common to both A - B and A - C. This result represents the right-hand side of our identity.
Verification
Comparing the left-hand side and the right-hand side, we have:
- A - (B ∪ C) = {4, 12}
- (A - B) ∩ (A - C) = {4, 12}
Since both sides are equal, we have successfully verified the identity:
A - (B ∪ C) = (A - B) ∩ (A - C)
This verification illustrates how the set difference operation interacts with union and intersection, providing a deeper understanding of set theory principles.
Identity (iii): A - (B ∩ C) = (A - B) ∪ (A - C)
The third identity explores another facet of set difference, this time in relation to the intersection of sets. The identity is expressed as:
A - (B ∩ C) = (A - B) ∪ (A - C)
This identity states that the set difference between A and the intersection of B and C is equivalent to the union of the set differences between A and B, and A and C. To verify this, we will compute both sides of the equation separately and demonstrate their equality, involving the intersection of B and C, set differences, and their union.
Left-Hand Side: A - (B ∩ C)
First, we need to find the intersection of sets B and C, denoted as B ∩ C. The intersection of two sets contains only the elements that are common to both sets. Given B = {1, 3, 5, 15} and C = {1, 2, 3, 6, 9, 18}, their intersection is:
B ∩ C = {1, 3}
This set represents the elements that are present in both B and C. Next, we find the set difference between A and this intersection. The set difference A - (B ∩ C) contains elements that are in A but not in B ∩ C. Given A = {1, 2, 3, 4, 6, 12} and B ∩ C = {1, 3}, the set difference is:
A - (B ∩ C) = {2, 4, 6, 12}
This set includes elements from A that are not present in B ∩ C. This result forms the left-hand side of our identity, which we will compare with the right-hand side.
Right-Hand Side: (A - B) ∪ (A - C)
We have already calculated the set differences A - B and A - C in the previous verification:
- A - B = {2, 4, 6, 12}
- A - C = {4, 12}
Now, we find the union of these two set differences. The union of A - B and A - C is:
(A - B) ∪ (A - C) = {2, 4, 6, 12} ∪ {4, 12} = {2, 4, 6, 12}
This set contains all unique elements from both A - B and A - C. This result represents the right-hand side of our identity.
Verification
Comparing the left-hand side and the right-hand side, we have:
- A - (B ∩ C) = {2, 4, 6, 12}
- (A - B) ∪ (A - C) = {2, 4, 6, 12}
Since both sides are equal, we have successfully verified the identity:
A - (B ∩ C) = (A - B) ∪ (A - C)
This verification highlights the interaction between set difference, intersection, and union, reinforcing the fundamental principles of set theory.
Identity (iv): A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
The fourth and final identity we will examine is another distributive law, this time for the union operation over intersection. The identity is stated as:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
This identity asserts that the union of set A with the intersection of sets B and C is equivalent to the intersection of the unions of A with B and A with C. To verify this, we will calculate both sides of the equation separately and demonstrate their equality. This involves finding the intersection of B and C, the unions of A with B and A with C, and the final union and intersection.
Left-Hand Side: A ∪ (B ∩ C)
We have already calculated the intersection of sets B and C in the previous verification:
B ∩ C = {1, 3}
Now, we need to find the union of set A with this intersection. The union of two sets contains all the elements present in either set. Given A = {1, 2, 3, 4, 6, 12} and B ∩ C = {1, 3}, their union is:
A ∪ (B ∩ C) = {1, 2, 3, 4, 6, 12}
This set includes all unique elements from both A and B ∩ C. This result forms the left-hand side of our identity, which we will compare with the right-hand side.
Right-Hand Side: (A ∪ B) ∩ (A ∪ C)
To compute the right-hand side, we first find the union of A and B, denoted as A ∪ B. Given A = {1, 2, 3, 4, 6, 12} and B = {1, 3, 5, 15}, their union is:
A ∪ B = {1, 2, 3, 4, 6, 12, 5, 15}
This set contains all unique elements from both A and B. Next, we find the union of A and C, denoted as A ∪ C. Given A = {1, 2, 3, 4, 6, 12} and C = {1, 2, 3, 6, 9, 18}, their union is:
A ∪ C = {1, 2, 3, 4, 6, 12, 9, 18}
This set includes all unique elements from both A and C. Finally, we find the intersection of these two unions. The intersection of A ∪ B and A ∪ C is:
(A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 5, 6, 12, 15} ∩ {1, 2, 3, 4, 6, 9, 12, 18} = {1, 2, 3, 4, 6, 12}
This set contains elements that are common to both A ∪ B and A ∪ C. This result represents the right-hand side of our identity.
Verification
Comparing the left-hand side and the right-hand side, we have:
- A ∪ (B ∩ C) = {1, 2, 3, 4, 6, 12}
- (A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 6, 12}
Since both sides are equal, we have successfully verified the identity:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
This verification further solidifies our understanding of set operations and their distributive properties, showcasing the elegance and consistency of set theory principles.
Conclusion
In this comprehensive exploration, we have meticulously verified four fundamental set theory identities using specific sets A, B, and C. Through detailed step-by-step calculations and explanations, we have demonstrated the validity of these identities. These verifications not only reinforce our understanding of set operations such as union, intersection, and difference but also highlight the distributive properties that govern their interactions. These principles are essential in various fields of mathematics, computer science, and beyond, providing a solid foundation for more advanced concepts and applications.
