Problem 12 - Discrete Mathematics with Applications

Question

Let \(A\) and \(B\) be finite sets such that \(A \subseteq B,|A|=b,|B|=a+b .\) Find the cardinality of each set. \(A \cup B\)

Step-by-Step Solution

Answer

The cardinality of the union of sets A and B, denoted as \(|A \cup B|\), is equal to the cardinality of set B. Since \( |B| = a + b\), the cardinality of A ∪ B is \(a + b\).

Step 1: Determine A is a subset of B

We are given that \(A \subseteq B\), meaning that all elements of set A are also in set B. Step 2: Analyze the Cardinalities

Step 2: Determine the cardinalities of A and B

The cardinalities of the sets A and B are given as follows: - |A| = b - |B| = a + b Step 3: Determine the Union Set Cardinality

Step 3: Determine the cardinality of A ∪ B

As A is a subset of B i.e. \(A \subseteq B\), all elements of set A are already in set B. So, the union of sets A and B will not have any additional elements from A. Thus, the cardinality of A ∪ B will be the same as the cardinality of B. So, we have \(|A \cup B| = |B|\). Step 4: Conclusion

Step 4: Find the cardinality of A ∪ B

Using the earlier determined relationship for the cardinalities, we have: \( |A \cup B| = |B|\) Since |B| = a + b, the required cardinality is: \( |A \cup B| = a + b\)

Key Concepts

Finite Sets

When we talk about finite sets in set theory, we are referring to sets that have a limited number of distinct elements. Unlike infinite sets, which go on endlessly, finite sets come with a specific count of members. This makes them easier to analyze and work with since their size, known as "cardinality," is well-defined.

For example, consider a set like \( A = \{1, 2, 3\} \). This set is finite because we can enumerate its elements, and more importantly, we can count them. In this case, there are three elements, so the cardinality \(|A|\) is 3. Finite sets are a foundational concept in mathematics, especially in problems where understanding the size and limits of a set is crucial.

Subset

In set theory, a subset is a set that contains only elements found in another set, which is referred to as the "superset." If set \( A \) is a subset of set \( B \), denoted as \( A \subseteq B \), all elements of \( A \) are also elements of \( B \). However, \( B \) might have additional elements not found in \( A \).

There are a few key points to remember about subsets:

• Every set is a subset of itself. Therefore, \( A \subseteq A \).
• The empty set is a subset of every set, regardless of the size or contents of the set.

In our original exercise, \( A \subseteq B \) indicates that all elements of \(A\) exist in \(B\), which helps in calculating the union and cardinality of these sets.

Cardinality

Cardinality is a concept that denotes the number of elements in a set. In simpler terms, it indicates the "size" of the set. For finite sets, the cardinality is a non-negative integer representing how many unique elements the set contains. Knowing the cardinality of a set is crucial when performing operations like union or intersection.

In our scenario, we are given the cardinality of two sets:

• \(|A| = b\)
• \(|B| = a + b\)

These values help us deduce information about how many elements are in these sets and their relationship. Understanding the cardinality concept allows us to determine aspects such as when two sets are equal or when one is larger than the other.

Union of Sets

The union of two sets is a fundamental operation in set theory that combines all elements from both sets, without repeating any elements. The union of sets \( A \) and \( B \) is denoted by \( A \cup B \).

For example, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), then \( A \cup B = \{1, 2, 3, 4, 5\} \). Notice that the element "3," which is present in both \( A \) and \( B \), appears only once in the union.

In our exercise, since \( A \subseteq B \), every element in \( A \) is already in \( B \). Therefore, the union \( A \cup B \) essentially recreates set \( B \), meaning the cardinality \(|A \cup B|\) is equal to \(|B|\), which is \(a + b\). This property simplifies calculations and is a vital technique in problems involving subsets and unions.