Finding The Cardinality Of Intersection Of Complements Of Sets

In the realm of set theory, understanding the relationships between sets and their complements is crucial. This article delves into a specific problem involving a universal set $U$ and its disjoint subsets $A$ and $B$. We are given the cardinality of the universal set, denoted as $n(U)$, and the cardinalities of the subsets $A$ and $B$, denoted as $n(A)$ and $n(B)$, respectively. Our goal is to determine the cardinality of the intersection of the complements of $A$ and $B$, which is represented as $n(A' \cap B')$. This exploration will not only provide a solution to this particular problem but also illuminate fundamental concepts in set theory, such as De Morgan's Laws and the principle of inclusion-exclusion.

Understanding the Problem

To effectively solve this problem, we must first dissect the given information and understand the notations used. We are given a universal set $U$, which encompasses all elements under consideration. The subsets $A$ and $B$ are disjoint, meaning they have no elements in common. This is a crucial piece of information as it simplifies our calculations. We are provided with the following cardinalities:

• $n(U) = 42$: The total number of elements in the universal set.
• $n(A) = 15$: The number of elements in subset $A$.
• $n(B) = 13$: The number of elements in subset $B$.

The notation $A'$ represents the complement of set $A$, which includes all elements in the universal set $U$ that are not in $A$. Similarly, $B'$ represents the complement of set $B$. The symbol $\cap$ denotes the intersection of two sets, which is the set containing all elements that are common to both sets. Therefore, $A' \cap B'$ represents the set of elements that are in both the complement of $A$ and the complement of $B$.

Our objective is to find $n(A' \cap B')$, which represents the number of elements in the intersection of the complements of $A$ and $B$. This involves understanding the relationships between sets, their complements, and the operations of union and intersection. We will leverage key concepts and theorems from set theory to arrive at the solution.

Applying De Morgan's Laws and Set Theory Principles

To find $n(A' \cap B')$, we can utilize De Morgan's Laws, a fundamental concept in set theory. De Morgan's Laws provide a relationship between the complements of unions and intersections of sets. The laws are stated as follows:

1. $(A \cup B)' = A' \cap B'$
2. $(A \cap B)' = A' \cup B'$

In our case, the first law is particularly relevant. It states that the complement of the union of two sets is equal to the intersection of their complements. Applying this law to our problem, we have:

$A' \cap B' = (A \cup B)'$

This means that the intersection of the complements of $A$ and $B$ is the same as the complement of the union of $A$ and $B$. This transformation is crucial because it allows us to express the quantity we want to find in terms of the union of $A$ and $B$, which is often easier to calculate.

Next, we need to find the cardinality of the union of $A$ and $B$, denoted as $n(A \cup B)$. Since $A$ and $B$ are disjoint sets, they have no elements in common. This simplifies the calculation of the cardinality of their union. For disjoint sets, the cardinality of the union is simply the sum of the cardinalities of the individual sets:

$n(A \cup B) = n(A) + n(B)$

Substituting the given values, we get:

$n(A \cup B) = 15 + 13 = 28$

Now that we have the cardinality of the union of $A$ and $B$, we can find the cardinality of its complement. The complement of a set contains all elements in the universal set that are not in the original set. Therefore, the cardinality of the complement of $A \cup B$ is the difference between the cardinality of the universal set and the cardinality of $A \cup B$:

$n((A \cup B)') = n(U) - n(A \cup B)$

Substituting the values we have:

$n((A \cup B)') = 42 - 28 = 14$

Finally, since we know that $A' \cap B' = (A \cup B)'$, we can conclude that:

$n(A' \cap B') = n((A \cup B)') = 14$

Therefore, the cardinality of the intersection of the complements of $A$ and $B$ is 14.

Step-by-Step Solution

To summarize, here's a step-by-step solution to the problem:

1. Identify the given information:
   • $n(U) = 42$
   • $n(A) = 15$
   • $n(B) = 13$
   • $A$ and $B$ are disjoint sets.
2. Apply De Morgan's Law:
   • $A' \cap B' = (A \cup B)'$
3. Calculate the cardinality of the union of A and B:
   • Since $A$ and $B$ are disjoint, $n(A \cup B) = n(A) + n(B) = 15 + 13 = 28$
4. Calculate the cardinality of the complement of the union:
   • $n((A \cup B)') = n(U) - n(A \cup B) = 42 - 28 = 14$
5. Conclude the result:
   • $n(A' \cap B') = n((A \cup B)') = 14$

This step-by-step approach breaks down the problem into smaller, manageable parts, making it easier to understand and solve. Each step utilizes a fundamental concept or theorem from set theory, leading us to the final answer.

Significance and Applications

The problem we've solved highlights the importance of set theory in various fields, including mathematics, computer science, and statistics. Understanding concepts like complements, unions, intersections, and De Morgan's Laws is crucial for working with data, logic, and probability.

In computer science, set theory is used in database management, algorithm design, and software engineering. For example, database queries often involve set operations like union, intersection, and difference to retrieve specific data. Algorithms may use set theory principles to optimize performance and ensure correctness.

In statistics, set theory is fundamental to probability theory. Events can be represented as sets, and the probability of events occurring together or separately can be calculated using set operations. Understanding set theory is essential for analyzing data and making informed decisions.

Furthermore, De Morgan's Laws have applications in logic and circuit design. They provide a way to simplify logical expressions and design more efficient circuits. By understanding these laws, engineers can create systems that are both reliable and cost-effective.

In conclusion, the problem of finding $n(A' \cap B')$ not only provides a practical exercise in set theory but also demonstrates the broader applicability of these concepts in various fields. By understanding the relationships between sets and their complements, we can solve complex problems and gain valuable insights in diverse areas of study and practice.

Conclusion

In summary, we have successfully determined $n(A' \cap B')$ by applying the principles of set theory, particularly De Morgan's Laws and the properties of disjoint sets. The problem involved finding the cardinality of the intersection of the complements of two disjoint subsets within a universal set. By leveraging the relationship $A' \cap B' = (A \cup B)'$ and calculating the cardinality of $(A \cup B)'$ using $n((A \cup B)') = n(U) - n(A \cup B)$, we arrived at the solution $n(A' \cap B') = 14$. This exercise not only reinforces our understanding of set theory concepts but also demonstrates their practical application in problem-solving. The systematic approach, involving breaking down the problem into smaller steps and applying relevant theorems, is a valuable skill in mathematics and beyond. Moreover, the significance of set theory in various fields, such as computer science and statistics, highlights the importance of mastering these fundamental concepts. Understanding the relationships between sets, their complements, and the operations that connect them allows us to approach complex problems with clarity and precision, making informed decisions based on logical reasoning and mathematical principles. This exploration serves as a testament to the power and versatility of set theory in unraveling mathematical intricacies and its relevance in a wide array of real-world applications.