Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A=\\{a, e, i, o, u\\}\) and \(B=\\{b, d, e, o, u\\}\). Verify by direct computation that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\).

Step-by-Step Solution

Answer

We have sets \(A = \{a, e, i, o, u\}\) and \(B = \{b, d, e, o, u\}\). By counting the elements, we find \(n(A) = 5\) and \(n(B) = 5\). The intersection \(A \cap B = \{e, o, u\}\) has 3 elements, so \(n(A \cap B) = 3\). The union of the sets is \(A \cup B = \{a, b, d, e, i, o, u\}\), which has 7 elements, giving \(n(A \cup B) = 7\). Substituting into the formula, we find that the formula \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\) holds true, since \(7 = 5 + 5 - 3\).

Step 1: Determine the number of elements in set A and B

First, we need to find the number of elements in sets A and B. We can do this by counting the elements in each set. For set A: \(A = \{a, e, i, o, u\}\) There are 5 elements in set A, so we have \(n(A) = 5\). For set B: \(B = \{b, d, e, o, u\}\) There are 5 elements in set B, so we have \(n(B) = 5\).

Step 2: Find the intersection of sets A and B

Next, we need to find the elements that are common between sets A and B, which is the intersection of the sets. To do that, we compare the elements in both sets and write down the common elements. \(A \cap B = \{e, o, u\}\)

Step 3: Determine the number of elements in the intersection

Now we need to count the number of elements in the intersection of sets A and B. \(n(A \cap B) = 3\) Since there are 3 elements in the intersection, we have \(n(A \cap B) = 3\).

Step 4: Find the union of sets A and B

Next, we need to find the union of sets A and B. This is the set that includes all unique elements from both sets, without repetition. \(A \cup B = \{a, b, d, e, i, o, u\}\)

Step 5: Determine the number of elements in the union

Now we need to count the number of elements in the union of sets A and B. \(n(A \cup B) = 7\) Since there are 7 elements in the union, we have \(n(A \cup B) = 7\).

Step 6: Verify the formula

Now that we have computed the number of elements in sets A, B, their intersection, and their union, we can substitute the values into the given formula: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\) \(7 = 5 + 5 - 3\) This simplifies to: \(7 = 7\) Since the formula holds true, we have successfully verified it by direct computation.

Key Concepts

Union of Sets

The concept of the union of sets is one of the foundational ideas in set theory. It refers to combining all the distinct elements from multiple sets into one single set, without duplicating any elements. When we talk about the union of sets, we use the symbol \(\cup\). Suppose we have two sets, \(A\) and \(B\). Their union, denoted as \(A \cup B\), would include every element that is in \(A\), \(B\), or both.

For example, if \(A = \{a, e, i, o, u\}\) and \(B = \{b, d, e, o, u\}\), the union \(A \cup B\) will be \(\{a, b, d, e, i, o, u\}\), since those are all the unique elements from both sets. Union is useful in various real-world contexts, such as combining different datasets, unifying groups, or merging resources.

Intersection of Sets

The intersection of sets can be thought of as the common ground between sets. This is where set theory intersects, literally! Represented by the symbol \(\cap\), the intersection of two sets \(A\) and \(B\), written as \(A \cap B\), is the set containing all elements that are both in \(A\) and in \(B\).

Using our example, for \(A = \{a, e, i, o, u\}\) and \(B = \{b, d, e, o, u\}\), the intersection \(A \cap B\) is \(\{e, o, u\}\), because these are the elements present in both sets. Understanding intersections is crucial in disciplines like logic, probability, and database management, as it helps identify shared characteristics or common data points.

Element Counting

Element counting, in the context of set theory, is a fundamental method for quantifying the contents of sets. By counting the number of unique elements, denoted as \(n(A)\) for set \(A\), we measure the cardinality of the set. Accurate element counting is essential; it's akin to taking inventory in a store to know what you have on hand.

When we deal with problems involving union and intersection, such as verifying \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\), we're essentially using element counting to prove set identities or solve equations. It’s critical to remember that items which appear in more than one set are only counted once in the union, but they are the essence of the intersection set count.

Venn Diagrams

Venn diagrams are the pictorial champions of set theory. They allow us to visualize the relationships between different sets, highlighting unions, intersections, and even disjoint sets. By drawing circles or other shapes to represent each set, we clearly see where they overlap—this is the intersection—and the areas that are unique to each or shared by multiple sets—the union.

Using our sets \(A\) and \(B\) again, we could draw two overlapping circles, label them, and place the elements accordingly. E.g., \(e\), \(o\), and \(u\) will be in the overlapping area to visualize \(A \cap B\), while the non-overlapping parts of the circles will contain the elements unique to each set. Venn diagrams are excellent educational tools, making abstract concepts tangible, particularly for visual learners.