Problem 40 - Precalculus Mathematics for Calculus

Question

Find the indicated set if $$\begin{array}{c}A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\} \\\C=\\{7,8,9,10\\}\end{array}$$ (a) \(B \cup C\) (b) \(B \cap C\)

Step-by-Step Solution

Answer

(a) \(B \cup C = \{2, 4, 6, 7, 8, 9, 10\}\), (b) \(B \cap C = \{8\}\).

Step 1: Understand Set Union

The union of two sets, denoted as \(B \cup C\), consists of elements that are in either set or in both. For this problem, \(B = \{2, 4, 6, 8\}\) and \(C = \{7, 8, 9, 10\}\).

Step 2: Find Elements in Union

To determine \(B \cup C\), list all unique elements from both sets \(B\) and \(C\). Thus, \(B \cup C = \{2, 4, 6, 7, 8, 9, 10\}\).

Step 3: Understand Set Intersection

The intersection of two sets, denoted as \(B \cap C\), consists of elements that are common to both sets. For this problem, find elements common to both \(B = \{2, 4, 6, 8\}\) and \(C = \{7, 8, 9, 10\}\).

Step 4: Find Elements in Intersection

Determine the common elements in both sets \(B\) and \(C\), which is only the element \(8\). Thus, \(B \cap C = \{8\}\).

Key Concepts

Understanding Set Union

In set theory, the concept of a set union is straightforward and valuable when combining datasets. Imagine two groups of numbers or objects, and you want to combine them into one larger group. This process is known as taking the union of two sets, denoted by the symbol \(\cup\). The union incorporates all the distinct elements from both groups.

For example, if we have sets \(B = \{2, 4, 6, 8\}\) and \(C = \{7, 8, 9, 10\}\), to find \(B \cup C\), we list every element from both \(B\) and \(C\) without repeating any element. Here, the resulting union is \(\{2, 4, 6, 7, 8, 9, 10\}\).

A few key points to remember about set union:

• Every element in the union is unique; duplicates are removed.
• Union is an inclusive operation, meaning elements from both initial sets are included.
• Order does not matter in a set.

This inclusive process is crucial for tasks involving combined datasets, such as database queries or merging lists.

Understanding Set Intersection

Set intersection is another basic yet essential concept in set theory. It focuses on identifying common elements between two sets. The intersection of sets is symbolized by \(\cap\) and comprises only those elements that exist in both sets simultaneously.

Consider the following example: let \(B = \{2, 4, 6, 8\}\) and \(C = \{7, 8, 9, 10\}\). To determine \(B \cap C\), we look for numbers that appear in both \(B\) and \(C\). The shared element here is \(8\), so the intersection is \(\{8\}\).

Important points about set intersection include:

• The intersection is more exclusive compared to the union, as it only includes shared elements.
• If no elements are common, the intersection is the empty set, denoted as \(\{\}\).
• Intersections are useful in filtering datasets to find common attributes.

This concept applies well to real-world problems, such as identifying shared characteristics in various datasets or solving logical problems.

Applying Mathematics Problem-Solving with Set Theory

Mathematics problem solving using set theory enhances understanding and provides structured approaches to complex problems. Set theory constructs like union and intersection help organize and dissect information effectively.

When solving problems using sets, it's essential to comprehend the rules and properties of these operations. For example, knowing that a union is an inclusive operation, while an intersection is exclusive, can guide the approach we take.

In practice:

• Break down the problem into smaller, manageable parts using sets.
• Apply the operation best suited to the problem's requirements.
• Double-check results by reviewing each set's elements to ensure accuracy.

Set theory in mathematics isn't just about numbers; it applies to logical grouping in various contexts, like data management and software engineering. This makes understanding union and intersection pivotal in diverse problem-solving scenarios.