Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \begin{array}{l} \text { \\{GM, Ford, Chrysler\\}, \\{Daimler-Benz, Volkswagen\\}, \\{Toy- }\\\ \text { ota, Nissan\\} } \end{array} $$

Step-by-Step Solution

Answer

The smallest possible set containing the given sets as subsets is \(\{GM, Ford, Chrysler, Daimler-Benz, Volkswagen, Toyota, Nissan\}\).

Step 1: The given sets are: 1. \(\{GM, Ford, Chrysler\}\) - American car manufacturers. 2. \(\{Daimler-Benz, Volkswagen\}\) - German car manufacturers. 3. \(\{Toyota, Nissan\}\) - Japanese car manufacturers. #Step 2: Merge the given sets into one set without repetitions#

In order to find the smallest possible set containing the given sets as subsets, we must merge the given sets into one set without repetitions of elements. So, combining the given sets, we get: \[ \{GM, Ford, Chrysler, Daimler-Benz, Volkswagen, Toyota, Nissan\} \] The merged set has no repeating elements, therefore it is the smallest possible set containing the given sets as subsets.

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Step 6: Conclude with the answer

The smallest possible set containing the given sets as subsets is \(\{GM, Ford, Chrysler, Daimler-Benz, Volkswagen, Toyota, Nissan\}\).

Key Concepts

Union of Sets

The union of sets is a fundamental concept in set theory that involves combining elements from multiple sets into a single set. This new set contains all the elements from the original sets, but each element is listed only once, without repetition.
To find the union, look at all sets involved and simply bring together every unique element. This method is like collecting various items from different groups and ensuring that none is repeated, giving you a comprehensive collection. In mathematical notation, the union of two sets \(A\) and \(B\) is represented as \(A \cup B\).
Consider if you have three sets, such as

• Set 1: \(\{GM, Ford, Chrysler\}\)
• Set 2: \(\{Daimler-Benz, Volkswagen\}\)
• Set 3: \(\{Toyota, Nissan\}\)

To find their union, gather all unique elements: \(\{GM, Ford, Chrysler, Daimler-Benz, Volkswagen, Toyota, Nissan\}\). Notice, there are no duplicates in the union, reflecting each element’s unique presence across the merged sets.

Subset

A subset is a set where all its elements are contained within another set. If set \(A\) is a subset of set \(B\), every element of \(A\) is also an element of \(B\). We denote this relationship with the symbol \(\subseteq\).
For example, consider set \(A = \{Toyota, Nissan\}\) and set \(B = \{GM, Ford, Chrysler, Toyota, Nissan\}\). Here, set \(A\) is a subset of set \(B\) because all elements of \(A\) can be found in \(B\).
Applying this idea to our task, each of the original sets given (such as \(\{GM, Ford, Chrysler\}\)) is a subset of the larger combined set \(\{GM, Ford, Chrysler, Daimler-Benz, Volkswagen, Toyota, Nissan\}\). This implies that each element from the smaller sets is represented in the overall union set, affirming their subset relationship.

Set Operations

Set operations are mathematical rules applied to sets to produce new sets from existing ones. They help us understand relationships and hierarchy between different sets.
Key set operations include:

• Union: Combining all elements from the given sets, as discussed earlier.
• Intersection: Finding common elements between sets. If two sets have no elements in common, their intersection is an empty set.
• Difference: Subtracting elements of one set from another, resulting in elements unique to the first set.
• Complement: Elements not present in the specified set, typically in relation to a universal set.

In the context of our example, the operation performed was a union, which effectively merged all elements to reach the smallest set containing all original sets as subsets. By understanding various set operations, especially union, we can resolve complex problems like determining minimal comprehensive sets, as seen in the exercise solution.