Problem 22 - Discrete Mathematics with Applications

Question

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h |, C=\\{c, d, f, g\\}, \text { and }\) \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. $$ A \cup(B \cap C) $$

Step-by-Step Solution

Answer

The binary representation of the set expression \(A \cup (B \cap C)\) is \(11101101\).

Step 1: Find the binary representation of sets A, B, C, and U

We have: Set A: \(\\{a, b, e, h\\}\) Set B: \(\\{b, c, e, f, h\\}\) Set C: \(\\{c, d, f, g\\}\) Universal Set U: \(\\{a, b, c, d, e, f, g, h\\}\) To find the binary representation of each set, we will assign '1' if an element is part of the set, and '0' if it is not part of the set. Using this method, we get the following binary representations: Set A: 11001001 Set B: 01101101 Set C: 00110110 Universal Set U: 11111111

Step 2: Calculate the intersection between sets B and C

The intersection of sets B and C contains the elements that are common to both sets: \(B \cap C = \\{c, f\\}\)

Step 3: Find the binary representation of the intersection of B and C

Using the method described in Step 1, we can find the binary representation of the intersection of sets B and C: \(B \cap C\): 00100100

Step 4: Calculate the union between set A and the intersection of B and C

To find the union of set A and the intersection of sets B and C, we combine elements from both sets: \(A \cup (B \cap C) = \\{a, b, c, e, f, h\\}\)

Step 5: Find the binary representation of the final result

Using the method described in Step 1, we can find the binary representation of the final result: \(A \cup (B \cap C)\): 11101101 The binary representation of the set expression \(A \cup (B \cap C)\) is \(11101101\).

Key Concepts

Binary Representation of Sets

In set theory, it's helpful to use binary representation to keep track of elements. By associating each element with either a '1' or a '0', you create a binary code that tells you what is included in the set.

For example, given a Universal Set \(U = \{a, b, c, d, e, f, g, h\}\), each element has a position. If a set contains an element, it is marked as '1'; otherwise, it is '0'. So for Set \(A = \{a, b, e, h\}\), the binary representation is \(11001001\). This method provides an easy and systematic way to represent and analyze sets.

Union and Intersection of Sets

The operations of union and intersection allow us to combine or find common elements in sets. Let's break these down:

- **Union** (\(A \cup B\)) combines all elements from both sets. If an element is in either set, it appears in the result.

- **Intersection** (\(A \cap B\)) shows elements common to both sets. Only elements that exist in both sets are included.

In the original exercise, the intersection of \(B = \{b, c, e, f, h\}\) and \(C = \{c, d, f, g\}\) gives \(B \cap C = \{c, f\}\). Later, the union of \(A\) and \(B \cap C\) combines these results to form \(\{a, b, c, e, f, h\}\).

Universal Set

The Universal Set, denoted as \(U\), contains all possible elements being considered in a given discussion. It acts as a backdrop for all other sets.

For instance, in our exercise, the universal set is \(U = \{a, b, c, d, e, f, g, h\}\). Every other set is a subset of \(U\). The universal set is useful because it helps define the frame of reference for all set operations and ensures that all elements are considered in the analysis.

Set Operations

Set operations like union, intersection, and complement are fundamental tools in set theory. They help create new sets from existing ones, analyze relationships, and find meaningful insights.

- **Union** merges sets to show all unique elements.

- **Intersection** focuses on commonalities.

- **Complement** finds elements not in a given set but present in the universal set.

These operations allow us to understand and manipulate sets easily, providing a clear way to solve problems, as seen in the exercise where set operations combine and intersect different sets.

Binary Operations in Sets

Binary operations in sets utilize the concept of '0s' and '1s' to perform operations like union and intersection effectively.

This approach helps clarify processes by simplifying calculations.

• **Intersection**: Perform a logical 'AND' operation. Elements present in both sets are marked as '1'.
• **Union**: Use a logical 'OR' operation. Any element present in either set gets a '1'.

These binary operations enable precise calculations, as demonstrated when you compute \(A \cup (B \cap C)\) to get \(11101101\) in binary. This clarity and precision make binary operations an essential tool in set theory.