Problem 36 - Algebra and Trigonometry

Question

\(35-38=\) Find the indicated set if \(A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\}\) \(C=\\{7,8,9,10\\}\) $$ \begin{array}{ll}{\text { (a) } B \cup C} & {\text { (b) } B \cap C}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding Union Operation

The union of two sets \(B\) and \(C\) is found by combining all the elements from both sets without repeating any elements. This operation identifies all unique elements that are either in \(B\), \(C\), or in both.

Step 2: Calculate Union of B and C

Set \(B = \{2, 4, 6, 8\}\) and Set \(C = \{7, 8, 9, 10\}\). To find \(B \cup C\), list all elements without duplicates: \(B \cup C = \{2, 4, 6, 8, 7, 9, 10\}\).

Step 3: Understanding Intersection Operation

The intersection of two sets \(B\) and \(C\) includes only the elements that are present in both sets. This operation finds common elements.

Step 4: Calculate Intersection of B and C

Both Sets \(B = \{2, 4, 6, 8\}\) and \(C = \{7, 8, 9, 10\}\) contain the element \(8\). Therefore, \(B \cap C = \{8\}\).

Step 5: Compile Results

From the union and intersection operations, we have found \(B \cup C = \{2, 4, 6, 8, 7, 9, 10\}\) and \(B \cap C = \{8\}\).

Key Concepts

union of sets

The union of sets is a fundamental concept in set theory, representing the combination of elements from two or more sets. For example, if you have two sets, say Set B = \( \{2, 4, 6, 8\} \) and Set C = \( \{7, 8, 9, 10\} \), the union of these sets, denoted as \( B \cup C \), includes all elements from both sets without any repetition.

To calculate the union:\

• Start by listing every element from both sets.
• If an element appears in both, like \(8\) does in our example, you only write it once.
• For \( B \cup C \), this gives us \( \{2, 4, 6, 8, 7, 9, 10\} \).

No element is duplicated in the union set, ensuring each member is unique. This concept is vital for efficiently gathering elements from multiple groups.
Understanding how to perform unions helps in various applications in mathematics and computer science, where combining datasets or information is essential.

intersection of sets

The intersection of sets focuses on identifying common elements shared between two or more sets. For instance, when considering the same Sets B = \( \{2, 4, 6, 8\} \) and C = \( \{7, 8, 9, 10\} \), the intersection is denoted as \( B \cap C \). This results in a set that includes only the elements present in both initial sets.

To determine the intersection:\

• Look for elements that appear in both sets.
• In our example, \(8\) is the only number present in both Set B and Set C.
• Thus, \( B \cap C = \{8\} \).

The concept of intersection is crucial as it allows us to find commonalities, providing insights into shared features or characteristics across different datasets.

set theory

Set theory is the mathematical study of collections of objects, called sets, which can be anything from numbers to more abstract concepts. The primary operations in set theory, such as union and intersection, allow us to manage and analyze these collections systematically.

• Union and intersection help us understand relationships and commonalities among different sets.
• Set theory forms the basis for various mathematical disciplines, including logic, statistics, and artificial intelligence.
• It introduces notations that simplify the language of mathematics, such as \( \cup \) for union and \( \cap \) for intersection.
• Understanding set theory is critical for deciphering more complex mathematical concepts and problems.

Set theory enables efficient problem-solving by providing structured methods to handle different collections and the operations that apply to them.