Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find \(n(A \cup B)\) given that \(n(A)=6, n(B)=10\), and \(n(A \cap B)=3 .\)

Step-by-Step Solution

Answer

The number of elements in the union of sets A and B is \(n(A \cup B) = n(A) + n(B) - n(A \cap B) = 6 + 10 - 3 = 13\).

Step 1: Write the given information

We are given: \(n(A)=6\), \(n(B)=10\), \(n(A \cap B)=3\).

Step 2: Apply the formula for finding the union of sets

The formula for finding the number of elements in the union of sets A and B is: \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\]

Step 3: Substitute the given values into the formula

Now, substitute the given values of \(n(A)\), \(n(B)\) and \(n(A \cap B)\) into the formula: \begin{align*} n(A \cup B) &= n(A) + n(B) - n(A \cap B) \\ &= 6 + 10 - 3 \end{align*}

Step 4: Calculate the result

Finally, calculate the value of \(n(A \cup B)\): \[n(A \cup B) = 6 + 10 - 3 = 13\] So, \(n(A \cup B) = 13\).

Key Concepts

Set Theory

Set theory forms the foundation for understanding how various collections of elements interact with each other. The concept of a set pertains to any well-defined collection of objects, which could be numbers, symbols, or even other sets. In the context of elementary set theory, common operations include the union, intersection, and difference of sets.

The union of two sets, denoted as \(A \cup B\), refers to a set containing all elements that are in A, B, or both. Essentially, it combines all unique items from both sets. The intersection of two sets, shown as \(A \cap B\), includes only the elements that are present in both sets. In some cases, understanding these basic operations can be crucial for solving problems involving sets, such as figuring out the total number of different items when two collections are combined.

Problem-solving

Using set theory for problem-solving involves analyzing the given information and systematically applying mathematical operations to find solutions. In a typical problem involving sets, it's important to:

• Identify and understand the elements within each set.
• Recognize the relationships between the sets, as indicated by operations such as unions or intersections.
• Apply the appropriate formulas to compute the desired results, such as the total elements in a union or those in an intersection.

When given a problem like finding \(n(A \cup B)\), it's crucial to follow a step-by-step approach. First, ensure you understand the terms involved and what is being asked of you. Then, methodically apply formulas while substituting known values. This structured approach simplifies the problem-solving process, making it less daunting and more manageable.

Mathematical Formula

At the heart of many set theory problems is the mathematical formula for the union of two sets. When you need to find the number of elements in the union of sets A and B, the formula is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

Here's how the formula works:

• \(n(A)\) and \(n(B)\) represent the numbers of elements in sets A and B, respectively.
• \(n(A \cap B)\) is the number of elements common to both sets, which are counted twice when \(n(A)\) and \(n(B)\) are added together.
• The formula subtracts \(n(A \cap B)\) to avoid this duplication, ensuring that each element is only counted once in the union.

Using this formula helps in accurately solving problems of set union, leading to precise and logical conclusions about the number of overall unique elements within combined sets.