Problem 12 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find the Union of Sets A and B

To find the union of two sets, we combine all the elements from both sets without repetition. The union of set A and B is denoted as \(A \cup B\). As given \(A = \{a, b, e, h\}\) and \(B = \{b, c, e, f, h\}\). The union of A and B is found by combining their elements: \(A \cup B = \{a, b, c, e, f, h\}\). Step 2: Binary Representation

Step 2: Binary Representation

To represent the set \(A \cup B = \{a, b, c, e, f, h\}\), we will use binary digits. For this, we will compare the set with the universal set \(U = \{a, b, c, d, e, f, g, h\}\). If an element is present in the set \(A \cup B\), we will represent it with a 1, and if it's absent, with a 0. Compare the sets element by element and represent the result in binary form: - a: Present (1) - b: Present (1) - c: Present (1) - d: Absent (0) - e: Present (1) - f: Present (1) - g: Absent (0) - h: Present (1) The binary representation of set \(A \cup B\) is 11101101.

Key Concepts

Union of Sets

In set theory, the union of sets is an operation that combines all elements from a collection of sets. It's important to note that in a union, we don't repeat any elements. Every element from each participating set appears only once in the resulting set.
For example, when finding the union of two sets like A and B, we gather every element from both sets. In the exercise, we calculated the union of sets \(A = \{a, b, e, h\}\) and \(B = \{b, c, e, f, h\}\) to get \(A \cup B = \{a, b, c, e, f, h\}\).

• "a," "b," "e," and "h" are shared elements and are included only once.
• Elements "c" and "f" come from set B and are unique to it, so they are also part of the union.

Unions are useful in different mathematical contexts and real-life situations. In computational terms, they can help in operations like merging datasets or comparing different groups of items without overlaps.

Binary Representation

In the context of sets and universal sets, binary representation is a technique used to represent the presence or absence of elements. Each element in a universal set is assigned a binary digit: '1' if the element is present in the specific set and '0' if the element is absent.
To practice, let's take our union set \(A \cup B = \{a, b, c, e, f, h\}\) and compare it to the universal set \(U = \{a, b, c, d, e, f, g, h\}\). As we go through each element in U:

• 'a' is present, thus it gets a '1.'
• 'b' is present, thus it gets a '1.'
• 'c' is present, thus it gets a '1.'
• 'd' is absent, thus it gets a '0.'
• 'e' is present, thus it gets a '1.'
• 'f' is present, thus it gets a '1.'
• 'g' is absent, thus it gets a '0.'
• 'h' is present, thus it gets a '1.'

Hence, the binary representation for \(A \cup B\) is 11101101. This approach allows us to handle and manipulate data effectively, especially in computing and digital logic.

Universal Set

The universal set, denoted here as \(U = \{a, b, c, d, e, f, g, h\}\), is an important concept in set theory. It encompasses all possible elements under consideration, forming a base or reference set for other subsets.
In practical terms, defining a universal set helps when you're dealing with multiple smaller sets and need a common ground for operations like unions, intersections, and complements. It acts as the "universe" of discourse, meaning all other sets you deal with are contained within this larger set.
When conducting operations like binary representation, the universal set is used as a reference point to check the presence or absence of elements from other sets. This standardizes operations and makes the manipulation of sets in mathematical or digital spaces more coherent and systematic.

• It sets limits and guidelines for various operations in set theory.
• Supports consistent representation of data.
• Facilitates the logical and organized handling of multiple subsets.

Understanding the universal set is crucial to apply these concepts efficiently in both mathematical and real-world applications.