Finding The Cardinality Of Intersecting Complements In Disjoint Sets A Set Theory Problem

In the realm of set theory, understanding the relationships and operations between sets is fundamental. One common task involves determining the cardinality (number of elements) of sets resulting from operations like union, intersection, and complement. This article delves into a specific problem involving disjoint subsets within a universal set, aiming to find the cardinality of the intersection of the complements of two sets. To solve this, we'll use key concepts such as De Morgan's Laws and the properties of disjoint sets. Understanding these concepts and their applications is crucial for various fields, including computer science, statistics, and logic.

Let's consider a universal set $U$ containing a total of 42 elements, denoted as $n(U) = 42$. Within this universal set, we have two disjoint subsets, $A$ and $B$. This means that $A$ and $B$ have no elements in common. The number of elements in set $A$ is 15, represented as $n(A) = 15$, and the number of elements in set $B$ is 13, represented as $n(B) = 13$. Our objective is to find the number of elements in the intersection of the complements of $A$ and $B$, which is written as $n(A' \cap B')$. This involves finding the elements that are neither in $A$ nor in $B$. The problem tests our understanding of set operations and how they relate to cardinality. By solving this problem, we reinforce our knowledge of De Morgan's Laws and the principles of inclusion and exclusion, which are essential tools in set theory and combinatorics. The concepts used here are applicable in various scenarios, such as database queries, where finding elements that satisfy multiple negative conditions is a common task.

Before diving into the solution, it's crucial to understand the underlying concepts. Firstly, the universal set $U$ is the set containing all possible elements under consideration. In our case, $U$ has 42 elements. Next, we have subsets $A$ and $B$. A subset is a set whose elements are all contained within another set. The term disjoint means that sets $A$ and $B$ have no elements in common; their intersection is an empty set, denoted as $A \cap B = \emptyset$. This implies that $n(A \cap B) = 0$. The complement of a set, denoted by $A'$ (or sometimes $A^c$), includes all elements in the universal set $U$ that are not in $A$. Similarly, $B'$ represents all elements in $U$ that are not in $B$. The intersection of two sets, $A' \cap B'$, consists of the elements that are present in both $A'$ and $B'$. De Morgan's Laws are particularly relevant here. The first law states that $(A \cup B)' = A' \cap B'$, meaning the complement of the union of $A$ and $B$ is equal to the intersection of their complements. The second law states that $(A \cap B)' = A' \cup B'$, meaning the complement of the intersection of $A$ and $B$ is equal to the union of their complements. These laws are fundamental in simplifying set operations and are widely used in logic and computer science for simplifying boolean expressions and optimizing database queries. Understanding these concepts provides the necessary foundation for solving the problem efficiently and accurately.

To find $n(A' \cap B')$, we can utilize De Morgan's First Law, which states that $(A \cup B)' = A' \cap B'$. Therefore, $n(A' \cap B') = n((A \cup B)')$. This transformation simplifies our task because we can now focus on finding the cardinality of the complement of the union of $A$ and $B$. The union of two sets, $A \cup B$, includes all elements that are in $A$, or in $B$, or in both. Since $A$ and $B$ are disjoint sets, their intersection is empty, meaning $A \cap B = \emptyset$. This simplifies the calculation of the cardinality of their union. The cardinality of the union of two sets is given by the formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. Since $A$ and $B$ are disjoint, $n(A \cap B) = 0$. Thus, the formula becomes $n(A \cup B) = n(A) + n(B)$. Substituting the given values, we have $n(A \cup B) = 15 + 13 = 28$. Now, to find the cardinality of the complement of the union, we use the formula: $n((A \cup B)') = n(U) - n(A \cup B)$. Substituting the values we have, $n((A \cup B)') = 42 - 28 = 14$. Therefore, $n(A' \cap B') = 14$. This means there are 14 elements in the universal set $U$ that are neither in set $A$ nor in set $B$. The solution demonstrates the power of using De Morgan's Laws to simplify set operations and highlights the importance of understanding the properties of disjoint sets. The steps involved are clear and logical, making the solution easy to follow and verify. This approach can be generalized to solve similar problems involving multiple sets and their complements.

Let's break down the solution into a step-by-step process to ensure clarity and understanding. First, we identify the given information: the cardinality of the universal set $n(U) = 42$, the cardinality of set $A$ $n(A) = 15$, and the cardinality of set $B$ $n(B) = 13$. Also, we know that sets $A$ and $B$ are disjoint, meaning they have no elements in common, so $n(A \cap B) = 0$. Our goal is to find $n(A' \cap B')$, the number of elements that are in the complements of both $A$ and $B$. Step 1 involves applying De Morgan's First Law, which states that $A' \cap B' = (A \cup B)'$. This allows us to rewrite our target as $n(A' \cap B') = n((A \cup B)')$. Step 2 is to find the cardinality of the union of $A$ and $B$. Since $A$ and $B$ are disjoint, we use the simplified formula: $n(A \cup B) = n(A) + n(B)$. Substituting the given values, we get $n(A \cup B) = 15 + 13 = 28$. Step 3 is to find the cardinality of the complement of the union. We use the formula: $n((A \cup B)') = n(U) - n(A \cup B)$. Substituting the values, we get $n((A \cup B)') = 42 - 28 = 14$. Step 4 is the final conclusion: Since $n(A' \cap B') = n((A \cup B)')$, we conclude that $n(A' \cap B') = 14$. This step-by-step breakdown illustrates how we systematically applied set theory principles and formulas to arrive at the solution. Each step is justified by a logical rule or a known formula, ensuring the correctness of the solution. The breakdown also serves as a guide for solving similar problems by providing a clear process to follow.

The concepts used in this problem have numerous practical applications across various fields. In computer science, set theory is fundamental to database management. For instance, when querying a database, operations like union, intersection, and complement are used extensively. Suppose we have a database of customers, and set $A$ represents customers who have purchased product X, and set $B$ represents customers who have purchased product Y. Finding $A' \cap B'$ would identify customers who have purchased neither product X nor product Y. This information can be valuable for targeted marketing campaigns or inventory management. In statistics, set theory is used in probability calculations. If we consider the universal set $U$ as the sample space, and $A$ and $B$ as events, then $A' \cap B'$ represents the event that neither $A$ nor $B$ occurs. The probability of this event can be calculated using the cardinality of the sets. In logic, set theory provides a foundation for propositional logic and predicate logic. De Morgan's Laws, in particular, have direct counterparts in logical equivalences, which are used to simplify logical expressions. For example, the logical equivalent of $(A \cup B)' = A' \cap B'$ is $\neg(P \lor Q) \equiv \neg P \land \neg Q$, where $P$ and $Q$ are logical propositions. In data analysis, set operations are used to filter and segment data. For example, if $A$ represents a set of customers with high engagement and $B$ represents customers with high spending, $A' \cap B'$ would identify customers who have low engagement but high spending, a segment that might require different marketing strategies. These examples highlight the broad applicability of set theory concepts in real-world scenarios, making the understanding of these concepts essential for professionals in various domains.

In conclusion, the problem of finding $n(A' \cap B')$ in a universal set with disjoint subsets $A$ and $B$ is a classic example of how set theory principles can be applied to solve practical problems. By understanding key concepts such as universal sets, subsets, disjoint sets, complements, and De Morgan's Laws, we were able to efficiently determine that $n(A' \cap B') = 14$. The step-by-step solution provided a clear and logical approach, demonstrating the importance of breaking down complex problems into manageable steps. The practical applications discussed highlight the relevance of set theory in various fields, including computer science, statistics, logic, and data analysis. Mastering these concepts not only enhances problem-solving skills but also provides a solid foundation for advanced topics in mathematics and related disciplines. The ability to manipulate sets and their operations is a valuable asset in many areas of professional and academic work. Therefore, a thorough understanding of set theory principles is highly beneficial for anyone working with data, logic, or mathematical models. This article aimed to provide a comprehensive explanation of the problem and its solution, emphasizing the underlying concepts and their applications, thereby fostering a deeper understanding of set theory and its practical implications.