Question

A car manufacturer determines that automatic transmission, power steering, and a CD player are the three most important features in generating sales. The production schedule for the next day has these features incorporated in cars as shown in the following table:$$\begin{array}{||c||c|c|c|}\hline \text { Car } & \text { Automatic Transmission } & \text { Power Steering } & \text { CD Player } \\\\\hline \text { A } & \mathrm{x} & \mathrm{x} & \\\\\hline \text { B } & \mathrm{x} & \mathrm{x} & \mathrm{x} \\\\\hline \text { C } & & \mathrm{x} & \\\\\hline \text { D } & & \mathrm{x} & \mathrm{x} \\\\\hline \text { E } & & & \mathrm{x} \\\\\hline \text {F} & & \mathrm{x} & \mathrm{x} \\\\\hline \text { G } & & & \mathrm{x} \\\\\hline \text {H} & & \mathrm{x} & \mathrm{x} \\\\\hline\end{array}$$ (a) How many cars have at least one of these features? Even though you can see the answer, use the Principle of Inclusion-Exclusion to derive it. (b) How many cars have two or more of these fcatures? Again, use the Principle of Inclusion-Exclusion to derive the answer.

Short answer

(a) 6 cars. (b) 5 cars.

Step-by-step solution

Define the Sets

Let's define three sets based on the features: \( A \) for cars with Automatic Transmission, \( P \) for cars with Power Steering, and \( C \) for cars with a CD Player. From the given table:\[ |A| = 2, \quad |P| = 5, \quad |C| = 5 \]

Determine Intersections of Two Sets

Identify how many cars have exactly two features by examining the table and calculating intersections:\[ |A \cap P| = 2, \quad |A \cap C| = 1, \quad |P \cap C| = 4 \]

Determine Intersection of Three Sets

Identify cars with all features:\[ |A \cap P \cap C| = 1 \]

Apply the Principle of Inclusion-Exclusion for Part (a)

To find cars with at least one feature, calculate using Inclusion-Exclusion:\[|A \cup P \cup C| = |A| + |P| + |C| - |A \cap P| - |A \cap C| - |P \cap C| + |A \cap P \cap C|\]\[= 2 + 5 + 5 - 2 - 1 - 4 + 1 = 6\]

Interpret Result of Part (a)

The calculation shows that 6 cars have at least one of the features.

Find Cars with Exactly Two Features for Part (b)

Use the set intersections found earlier:\[ (|A \cap P| - |A \cap P \cap C|) + (|A \cap C| - |A \cap P \cap C|) + (|P \cap C| - |A \cap P \cap C|)\]\[ = (2 - 1) + (1 - 1) + (4 - 1) = 1 + 0 + 3 = 4 \]

Find Cars with All Three Features for Part (b)

Add cars with all three features, which were previously calculated:\[ |A \cap P \cap C| = 1 \]

Find Total Cars with Two or More Features for Part (b)

Sum results from steps 6 and 7 for total cars with two or more features:\[ 4 + 1 = 5 \]

Interpret Result of Part (b)

The calculation shows that 5 cars have two or more features.

Key concepts

Set Theory

Set theory is a fundamental part of mathematics, and it is essentially the study of well-defined collections of objects, which we call sets. In our case, we are dealing with sets like \(A\) for cars with Automatic Transmission, \(P\) for cars with Power Steering, and \(C\) for cars with a CD Player. Understanding how these sets interact is crucial.
**Key Concepts in Set Theory:**- **Union (\(A \cup B\))**: This is the set of elements that are in either set \(A\) or set \(B\) or in both.- **Intersection (\(A \cap B\))**: This is the set of elements common to both sets \(A\) and \(B\).- **Cardinality**: This refers to the number of elements in a set, such as \(|A|\), which is the size of set \(A\).
For example, in the exercise, to determine the total number of cars with at least one feature, we use the union of the three sets \(A\), \(P\), and \(C\). We are looking for the cardinality of this union, expressed as \(|A \cup P \cup C|\). This requires proper understanding of both intersections and unions of sets.

Combinatorics

Combinatorics is the field of mathematics that concerns itself with counting, arrangement, and combination of objects. For this exercise, combinatorics helps us systematically count cars with different feature combinations.
**Using Combinatorics to Count Features:**- **Counting Sets**: Knowing \(|A| = 2\), \(|P| = 5\), and \(|C| = 5\), we can easily understand how many cars have each feature individually.- **Calculating Intersections**: We determined that \(|A \cap P| = 2\), \(|A \cap C| = 1\), and \(|P \cap C| = 4\), helping us know how many cars share two features.- **Inclusion-Exclusion Principle**: This combinatoric principle is used to calculate the number of elements in the union of multiple sets, taking into account overlaps to avoid double counting.
In our problem, to find cars with at least one feature, we summed up the individual numbers, subtracted the double counts (intersections of two), and added back the triple counts. This structured counting method ensures clarity and accuracy.

Mathematical Problem Solving

Mathematical problem solving involves using logic and structured methods to arrive at a solution. In this exercise, the steps demonstrate a systematic approach to solving a problem involving set theory and combinatorics.
**Steps in Problem Solving:** - **Define the Problem**: Clearly identify what needs to be calculated. For instance, outline which features and combinations are involved. - **Break Down into Steps**: Approach the problem incrementally. Start with simple counting (number of cars with each feature), then move on to combinations (intersections, unions). - **Apply Theorems/Principles**: Use applicable mathematical theories such as the Principle of Inclusion-Exclusion, allowing us to resolve multi-set problems effectively. - **Verify and Interpret Results**: After calculation, ensure the results make sense. For the exercise, checking the results for at least one feature and two or more features confirms our methods are correct.
Good problem solving is methodical and clear. It involves not just directly finding an answer but also understanding the relationships between different elements of a problem. This skill is key in mathematics and beyond.