Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

If \(n(A)=4, n(B)=5\), and \(n(A \cup B)=9\), find \(n(A \cap B)\).

Step-by-Step Solution

Answer

The number of elements in the intersection of sets A and B, denoted as \(n(A \cap B)\), is 0.

Step 1: Identify given information

We are given three pieces of information about sets A and B: 1. The number of elements in set A is 4, denoted as \(n(A) = 4\). 2. The number of elements in set B is 5, denoted as \(n(B) = 5\). 3. The number of elements in the union of A and B is 9, denoted as \(n(A \cup B) = 9\).

Step 2: Apply the Principle of Inclusion and Exclusion

The Principle of Inclusion and Exclusion states that: \[n(A \cup B) = n(A) + n(B) - n(A \cap B)\] Where \(n(A \cap B)\) is the number of elements in the intersection of A and B. Our goal is to find the value of \(n(A \cap B)\).

Step 3: Substitute the given information

Using the given information, substitute the values of \(n(A)\), \(n(B)\), and \(n(A \cup B)\) into the equation: \[9 = 4 + 5 - n(A \cap B)\]

Step 4: Solve for n(A ∩ B)

Now, solve for \(n(A \cap B)\): \[9 = 9 - n(A \cap B)\] Subtract 9 from both sides of the equation: \[0 = -n(A \cap B)\] Multiply both sides by -1: \[0 = n(A \cap B)\] So, the number of elements in the intersection of sets A and B, denoted as \(n(A \cap B)\), is 0.

Key Concepts

Set Theory

Set theory is the mathematical study of collections of distinct objects, which are called sets. Imagine sets as boxes filled with certain items where each item represents an element of the set. Actually, sets can contain anything from numbers and symbols to colors and shapes. A set can have a finite number of elements, like the number of students in a classroom, or it can be infinite, like the set of all even numbers.

Key concepts within set theory include the notation of sets, typically capitalized letters such as 'A' and 'B', and the size or cardinality of a set, represented as 'n(A)' for the number of elements in set A. Understanding set theory is crucial for topics across mathematics and science because it allows for the organization and manipulation of collections of objects in a clear and structured way.

Intersection of Sets

The intersection of sets refers to a new set containing all the elements that are common to both sets. Think of it like the overlap between two circles in a Venn diagram — this overlapping region represents the intersection. In mathematical terms, the intersection of two sets A and B is denoted by 'A ∩ B'.

If we take two sets of numbers, for instance, set A having 1, 2, 3, and 4, and set B having 3, 4, 5, and 6, the intersection 'A ∩ B' would be the numbers 3 and 4, since these are the only numbers that are in both sets. It's important to note that if two sets have no common elements, their intersection is called an 'empty set', denoted by the symbol '∅' or '{}'. This happens in our exercise, where the intersection of sets A and B is found to be 0, indicating that there are no common elements, despite both sets having elements of their own.

Union of Sets

The union of sets is a concept that combines all elements from both sets without duplication. It's like pouring two different boxes of legos into one bigger box, where you only keep one piece of each unique color or shape. Mathematically, the union of two sets A and B is denoted by 'A ∪ B'.

Using the same earlier sets, set A with elements 1, 2, 3, and 4, and set B with elements 3, 4, 5, and 6, the union 'A ∪ B' would include all the elements from both sets but listed only once. So 'A ∪ B' is 1, 2, 3, 4, 5, 6. The exercise given shows that the union of sets A and B was fully enumerated by 9 unique elements, demonstrating that although each set had its own members, not a single element was common to both, as deduced by applying the Principle of Inclusion and Exclusion.