Problem 39 - 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) \(A \cup B\) (b) \(A \cap B\)

Step-by-Step Solution

Answer

(a) \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\) (b) \(A \cap B = \{2, 4, 6\}\)

Step 1: Understand Union of Sets

The union of two sets \(A\) and \(B\), written as \(A \cup B\), includes all elements that are in either set \(A\) or set \(B\) or in both. In other words, it combines the elements of both sets without duplication.

Step 2: Find A Union B

To find \(A \cup B\), list all unique elements from both sets \(A\) and \(B\). Set \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and set \(B = \{2, 4, 6, 8\}\). Combine the elements: \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\).

Step 3: Understand Intersection of Sets

The intersection of two sets \(A\) and \(B\), written as \(A \cap B\), includes the elements that are common to both sets. This means we find all elements that are present in both \(A\) and \(B\) simultaneously.

Step 4: Find A Intersection B

To find \(A \cap B\), identify the elements that are present in both sets \(A\) and \(B\). Both sets contain the elements \(2, 4,\) and \(6\). Thus, \(A \cap B = \{2, 4, 6\}\).

Key Concepts

Union of Sets

The union of sets is a fundamental idea in set theory that combines elements from multiple sets without any repetition. When we talk about the union of two sets \(A\) and \(B\), denoted as \(A \cup B\), we include every element that appears in at least one of the sets. This means each element of set \(A\), each element of set \(B\), and any elements that appear in both are all counted in the union.

For instance, given two sets, \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and \(B = \{2, 4, 6, 8\}\), the union \(A \cup B\) combines all unique elements as follows:

• Elements from \(A\): 1, 2, 3, 4, 5, 6, 7
• Elements from \(B\): 2, 4, 6, 8

After listing all elements, \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\). Notice how each element is only listed once, even if it appears in both sets. Union helps in creating a comprehensive list without duplicates. This operation is useful for finding total members when categorizing data into overlapping groups.

Intersection of Sets

The intersection of sets is another critical operation in set theory. It identifies and collects only those elements that appear in both sets. When we take the intersection of two sets \(A\) and \(B\), denoted as \(A \cap B\), we focus on finding elements that are shared by both sets.

Let's take an example: \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and \(B = \{2, 4, 6, 8\}\). Here's how to find the intersection:

• Determine which elements are common in both sets. Here, the elements 2, 4, and 6 are present in both \(A\) and \(B\).
• Therefore, the intersection \(A \cap B = \{2, 4, 6\}\).

Intersection helps focus on commonalities, making it a powerful tool for identifying shared characteristics between groups. In real-world scenarios, intersections can apply to data analysis, where filtering for common attributes is essential.

Mathematical Notation

Mathematical notation is a language all its own. It helps to express complex mathematical ideas in a clear, concise way. In set theory, we use specific symbols to denote operations and relationships between sets.

Here are some key notations:

• Union: The union of sets \(A\) and \(B\) is written as \(A \cup B\). This symbol \(\cup\) signifies combining all unique elements from both sets.
• Intersection: The intersection is represented by \(A \cap B\). The symbol \(\cap\) indicates finding common elements shared between sets.
• Elements of a Set: Sets themselves are typically written in curly brackets, like \(\{a, b, c\}\), signifying a collection of items.

Using mathematical notation like this allows us to simplify and organize mathematical expressions, ensuring that communication of ideas remains clear and precise. Understanding and using these symbols effectively is crucial for solving set theory problems and beyond, as it forms the basis for mathematical reasoning and theoretical application.