Problem 15 - 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. \(A \cap B\)

Step-by-Step Solution

Answer

The cardinality of the intersection \(A \cap B\) is equal to the cardinality of A, which is \(b\).

Step 1: Understand the given information

We know that A ⊆ B, and given that |A| = b and |B| = a + b.

Step 2: Determine the intersection of A and B

Since A is a subset of B, every element of A is also an element of B. Therefore, the intersection of A and B (A ∩ B) contains all the elements that are common to both A and B. In this case, all elements of A are also elements of B, so A ∩ B = A.

Step 3: Find the cardinality of A ∩ B

Since A ∩ B = A, the cardinality of A ∩ B will be equal to the cardinality of A. From the given information, we know that |A| = b.

Step 4: Write the final answer

The cardinality of A ∩ B is equal to |A|, which is b.

Key Concepts

Set Intersection

Let's dig into what set intersection means. When we talk about the intersection of sets, we're referring to the elements that these sets have in common. If you have two sets, say set \(A\) and set \(B\), then their intersection, denoted as \(A\cap B\), consists of all the elements that are found in both set \(A\) and set \(B\).

In the exercise we're discussing, \(A\) is a subset of \(B\). This means every element of \(A\) can already be found in \(B\). It's like saying if \(A\) represents your favorite fruits and \(B\) is all fruits in your fridge, then \(A\cap B\) would just be your favorite fruits, as they're all present in the fridge. So here, \(A\cap B = A\).

Understanding intersections helps us see commonalities or shared parts between different sets in a simple manner. This fundamental concept is crucial in various fields like mathematics, computer science, and logic.

Set Cardinality

Cardinality is a fancy term for counting—specifically, counting the number of elements in a set. When determining the size of a set, mathematically, we use bars, such as \(|A|\), to denote the number of elements in set \(A\).

In our example, we know that \(|A| = b\). This means that there are \(b\) elements in set \(A\). Similarly, since the cardinality of \(B\) is given as \(|B| = a + b\), it implies there are \(a + b\) elements in set \(B\).

Counting the size of a set is important, as it helps to quantify and understand the scope of the elements within. It provides clarity on how large or small a set is relative to others, which is vital in analysis, probability, and data management.

Subset

Subset is another key concept when dealing with sets. If set \(A\) is called a subset of set \(B\), denoted \(A \subseteq B\), every element in \(A\) is also found in \(B\). This is akin to saying that all the apples in your basket (\(A\)) are within the larger collection of fruits you have (\(B\)).

In the problem at hand, \(A\) being a subset of \(B\) directly impacts our understanding of their intersection \(A \cap B\). As mentioned earlier, this intersection means every element of \(A\) is already included in \(B\), thus \(A \cap B\) essentially equals \(A\).

Subsets indicate the inclusion relationship between sets and serve as an essential concept for discussions about set hierarchy or organization, whether you're arranging data, constructing logic models, or analyzing groups of information.