Problem 21 - 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. $$ \\{1,2\\},\\{1,3,4\\},\\{4,6,8,10\\} $$

Step-by-Step Solution

Answer

The smallest possible set containing the given sets as subsets is \( \{1, 2, 3, 4, 6, 8, 10\} \).

Step 1: Identify the elements in each given set

The given sets are: 1. \( \{1, 2\} \) 2. \( \{1, 3, 4\} \) 3. \( \{4, 6, 8, 10\} \)

Step 2: Merge the elements of all the given sets

Combine all elements in the given sets without duplications: \( \{1, 2, 3, 4, 6, 8, 10\} \)

Step 3: Verify that each given set is a subset of the merged set

We need to ensure that each given set is a subset of the merged set. Each element of the given set should be present in the merged set. 1. \( \{1, 2\} \subseteq \{1, 2, 3, 4, 6, 8, 10\} \) 2. \( \{1, 3, 4\} \subseteq \{1, 2, 3, 4, 6, 8, 10\} \) 3. \( \{4, 6, 8, 10\} \subseteq \{1, 2, 3, 4, 6, 8, 10\} \) As all of the given sets are subsets of the merged set, we have found the smallest possible set containing the given sets as subsets.

Step 4: Write down the final answer

The smallest possible set containing the given sets as subsets is \( \{1, 2, 3, 4, 6, 8, 10\} \).

Key Concepts

Subsets

In set theory, a subset is a set that contains some or all elements of another set. Think of a subset as a smaller group pulled from a larger group.
For example, if we have a set \( A = \{1, 2, 3\} \), then \( \{1, 2\} \) and \( \{3\} \) are subsets of \( A \).
A subset can even be the full set itself. In this exercise, identifying subsets helps us confirm whether each of the given sets fits into our merged set. Remember these key points about subsets:

• A subset can match the original set completely, or it can have fewer elements.
• The empty set, \( \{\} \), is a subset of every set.
• We denote the subset relationship using the symbol \( \subseteq \), which means "is a subset of".

This concept ensures we can verify that one set is contained within another, which is central to solving problems asking for smallest sets containing others as subsets.

Union of Sets

The union of sets is a critical operation in set theory, where all elements from multiple sets are combined to form a new set containing all unique elements from these original sets. In this operation, duplicates are removed. For example, if we have two sets \( A = \{1, 2\} \) and \( B = \{2, 3\} \), their union is \( A \cup B = \{1, 2, 3\} \). In the exercise, we merged the given sets:

• \( \{1, 2\} \)
• \( \{1, 3, 4\} \)
• \( \{4, 6, 8, 10\} \)

The union operation collects all different elements into a single set, \( \{1, 2, 3, 4, 6, 8, 10\} \). This is the smallest set that can contain all given sets as subsets. Key points about unions:

• The union combines all elements, but only once.
• Use the symbol \( \cup \) to denote union.
• Order does not matter; \( A \cup B = B \cup A \).

This fundamental operation aids in solving the problem by ensuring all elements from each set are considered.

Mathematics Problem Solving

Solving problems in mathematics involves several key strategies, which help break down complex tasks into manageable steps. Let's explore these strategies using the set theory problem we tackled.
Begin by understanding the question: What are we trying to achieve? In this exercise, the goal was to find the smallest set that contains each provided set as a subset. The steps involved include:

• Identify elements in each set to ensure completeness.
• Combine all elements through the union, ensuring we only keep unique items.
• Verify that each original set is a subset of the resulting union set.

While problem-solving, these steps help ensure a clear direction:

• Break down the problem as much as possible.
• Think logically about the relations between different parts, like how subsets relate to supersets.
• Check and verify your results to ensure accuracy.

Understanding and applying systematic approaches can simplify complex problems and reveal efficient solutions, just like finding the smallest subset in this exercise.