Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A=\\{2,4,6,8\\}\) and \(B=\\{6,7,8,9,10\\} .\) Compute: a. \(n(A)\) b. \(n(B)\) c. \(n(A \cup B)\) d. \(n(A \cap B)\)

Step-by-Step Solution

Answer

a. \(n(A) = 4\) b. \(n(B) = 5\) c. \(n(A \cup B) = 7\) d. \(n(A \cap B) = 2\)

Step 1: a. Finding n(A)

To find the number of elements in set A, simply count the elements in the set. In this case, the elements of set A are {2, 4, 6, 8}. So, n(A) = 4.

Step 2: b. Finding n(B)

To find the number of elements in set B, count the elements in the set. In this case, the elements of set B are {6, 7, 8, 9, 10}. So, n(B) = 5.

Step 3: c. Finding n(A ∪ B)

To find the number of elements in the union of set A and B, we need to combine both sets, but only count distinct elements once. To do this, first list all the elements from set A and set B together: {2, 4, 6, 8, 6, 7, 8, 9, 10}. Now, remove the duplicates and count the remaining elements: {2, 4, 6, 8, 7, 9, 10}. So, n(A ∪ B) = 7.

Step 4: d. Finding n(A ∩ B)

To find the number of elements in the intersection of set A and B, we need to find the elements that are common to both sets. Compare the elements of set A and set B and find the elements that are in both: {6, 8}. So, n(A ∩ B) = 2.

Key Concepts

Number of Elements in a Set

Understanding the number of elements in a set, often denoted as 'n(A)' for set A, is foundational in set theory. To determine this, count each distinct element in the set. For example, if set A contains the elements \(\{2,4,6,8\}\), the count or 'cardinality' of A is 4, hence \(n(A)=4\). It is important to note that repetition of elements in a set does not increase its cardinality; a set is defined by unique elements only.

When learning about sets, start by listing the elements, as seeing them visually can help prevent confusion. It's also beneficial to practice with different types of sets, including empty sets (no elements), finite sets (countable number of elements), and infinite sets (uncountable elements). By regularly practicing counting elements, you'll become more familiar with set sizes and complexities.

Union of Sets

The union of two sets, represented by \(A \cup B\), is a fundamental operation in set theory. It combines all elements from both sets, omitting duplicates to maintain the definition of a set as a collection of unique elements.

To find the union of sets A and B, you write down all the elements from both sets and remove any repetitions. This can be visualized with a Venn diagram, where two overlapping circles represent the sets, and their union is the area covered by both circles combined. For the given sets A and B in our original example, the union set is \(\{2,4,6,8,7,9,10\}\), leading to \(n(A \cup B) = 7\) elements. Remember, the union is about inclusivity—every element from both sets gets a spot in the union set.

Intersection of Sets

The intersection of sets is denoted by \(A \cap B\), and it includes only those elements that are common to both sets A and B. This operation is akin to finding a common ground between two groups.

As an example, if set A is \(\{2,4,6,8\}\) and set B is \(\{6,7,8,9,10\}\), the intersection, \(A \cap B\), will have elements that appear in both, which are 6 and 8. Therefore, \(n(A \cap B) = 2\). Visual learners might find it helpful to depict this using a Venn diagram too, where the intersection is represented by the overlapping part of the circles. This shared area helps emphasize the commonality. Always focus on just the shared elements when finding intersections, as this is a key step to correctly understanding and analyzing relationships between data sets.