Problem 17 - Discrete Mathematics with Applications

Question

Let \(A\) and \(B\) be finite disjoint sets, where \(|A|=a,\) and \(|B|=b .\) Find the cardinality of each set. \(A-B\)

Step-by-Step Solution

Answer

Since A and B are finite disjoint sets with cardinalities a and b respectively, the set difference A-B includes all elements in A that are not in B. Since there are no common elements between disjoint sets A and B, the cardinality of A-B is the same as the cardinality of A. Therefore, the cardinality of A-B is \(a\).

Step 1: Identify the cardinality of A and B

Since given |-A| = a and |-B| = b, we know the number of elements in A is a and the number of elements in B is b.

Step 2: Determine the set difference A-B

Calculate the set difference A-B, which includes all elements in A that are not in B.

Step 3: Calculate the cardinality of A-B

As A and B are disjoint sets, there are no common elements between them. Therefore, the cardinality of A-B is the same as the cardinality of A. So, the cardinality of A-B is a.

Key Concepts

Understanding Finite Disjoint Sets

When we talk about finite disjoint sets, we're referring to two or more sets that have a limited number of elements (that's what finite means) and share no elements in common (that's the disjoint part). Imagine two circles that don't overlap at all; that's a good visual for disjoint sets.

Familiarizing ourselves with this concept is crucial when solving problems related to combining or comparing different groups. To put it simply, if you know two sets are disjoint, you can immediately conclude that they have nothing in common which greatly simplifies the process of analyzing the relationships between the sets.

Set Difference Explained

The set difference, represented as \( A - B \), is a fundamental concept in set theory. It refers to the elements that belong to one set, but not to another. To find \( A - B \) you would start with all the elements in set \( A \) and then 'subtract' any that are also in set \( B \). It's like taking a complete list of one group, and crossing out any names that also appear in another group.

What it boils down to, is that the set difference tells us what's unique to the first set. It's important to remember that \( A - B \) is not necessarily the same as \( B - A \) since each operation might yield a different set based on the unique elements of each original set.

Calculating Cardinality

Cardinality refers to the number of elements in a set. In the realm of finite sets, calculating cardinality is like counting the items in a collection. But it's not always as straightforward as it seems because you have to be mindful of duplicates and the relationship between the sets in question.

For instance, if you have two separate bags of fruits with no fruit in common (our disjoint sets), simply adding up the total number of fruits gives you the combined cardinality. This is because every fruit can only be counted once, as they are distinct. The scenario illustrated in the exercise, where sets \( A \) and \( B \) are disjoint, shows that the cardinality of \( A - B \) is simply the cardinality of \( A \) since by definition, \( B \) cannot take away any elements from \( A \) in the set difference operation.