Set Theory Explained: Finding ( n(A cap B') ) When Given ( n(U) ), ( n(A) ), ( n(B) ), and ( n(A cap B) )

Hello, and welcome to this guide on set theory, focusing on how to determine the intersection of a set and its complement when given certain conditions. If you’re a student of mathematics or a professional in fields like computer science, understanding these concepts can be incredibly useful. This article will step you through the problem with a given set of values and explain the mathematical reasoning behind each step.

Given:

( n(U) 80 ) ( n(A) 48 ) ( n(B) 25 ) ( n(A cap B) 25 )

Retailing the Problem: Finding ( n(A cap B') )

The goal is to find ( n(A cap B') ), where ( B' ) denotes the complement of ( B ).

Step-by-Step Solution:

Total Population: Given the total number of elements ( U ) as 80, this represents the total universe or set that contains all elements under consideration. Intersection of A and B: The number of elements in the intersection of A and B, denoted as ( A cap B ), is given as 25. This means that there are 25 elements that belong to both sets A and B. Determining the Complement of B: To find ( n(A cap B') ), we need to understand that ( B' ) (the complement of B) contains all elements in the universe ( U ) that are not in B. Since ( n(B) 25 ), the number of elements in ( B' ) is ( n(U) - n(B) ), which is 80 - 25 55. Applying the Principle of Complement: ( n(A cap B') ) can be found by subtracting the number of elements in ( A cap B ) from the number of elements in A, i.e., ( n(A) - n(A cap B) ). This is because the elements in ( A cap B ) are the elements in both A and B, and we want only those in A that are not in B (i.e., the complement of B in A).

Final Calculation:

[ n(A cap B') n(A) - n(A cap B) 80 - 25 55 ]

Understanding the Concepts:

Set Theory Basics:

Universal Set (U): The set that contains all elements under consideration. Intersection (( cap )): The intersection of two sets A and B (denoted as ( A cap B )) is the set of elements common to both A and B. Complement (( '' )): The complement of a set B, denoted as ( B' ), contains all elements in the universal set U that are not in B. Venn Diagrams: A visual representation that helps in understanding the relationships between sets, where the intersection and complement can be easily visualized.

Applications:

Set theory and its operations are widely used in various fields, including:

Computer Science: For data structures, algorithms, and database management systems. Mathematics: In algebra, probability, and statistics. Information Retrieval: For ranking and optimizing search results in websites and applications.

Conclusion:

By understanding the principles of set theory, you can solve problems involving intersections and complements. In this article, we demonstrated how to find ( n(A cap B') ) given certain values. This knowledge can be applied to a wide range of problems in academic and professional settings. If you have any more questions or need further clarification, feel free to reach out!