Problem 14 - 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. $$ A-B $$

Step-by-Step Solution

Answer

The binary representation of the set A-B is \(A-B = [1, 0, 0, 0, 0, 0, 0, 0]\).

Step 1: Find the relative complement of set B in set A (A-B)

To find A-B, we list down all the elements that are present in set A but not in set B. Given sets are: \(A = \{a, b, e, h\}\) \(B = \{b, c, e, f, h\}\) So, \(A-B = \{a\}\) since only 'a' is present in set A but not in set B.

Step 2: Find the binary representation of set A-B

To find the binary representation of A-B, we will use the universal set U. The binary representation of a set contains 1 if the element from the universal set is present in the set, and 0 if the element is not present. The universal set U is given by: \(U = \{a, b, c, d, e, f, g, h\}\) Since A-B contains only 'a,' it can be represented as: \(A-B = [1, 0, 0, 0, 0, 0, 0, 0]\) Now we have the binary representation of the set A-B.

Key Concepts

Binary Representation of Sets

Understanding the binary representation of sets can simplify and visualize operations in set theory. It involves expressing a set using a sequence of binary values based on a universal set. Here's how it works:

• A binary value of 1 indicates the presence of an element from the universal set in the given set.
• A binary value of 0 indicates the absence of that element in the given set.

To convert a set, like set A-B, into binary form, align each element of the universal set, U, with the values 1 or 0 depending on its presence in A-B. For example, given the universal set\( U = \{a, b, c, d, e, f, g, h\} \) and the result set \( A-B = \{a\} \), the binary sequence becomes \([1, 0, 0, 0, 0, 0, 0, 0]\). The '1' in the first position represents the inclusion of 'a', while the rest are '0's because no other elements from U belong to A-B.

Universal Set

A universal set is a comprehensive set that includes all objects under consideration in a particular discussion. It is denoted by the symbol U. In set operations, the universal set serves as a baseline to compare and analyze other sets.

For example, if we have sets A, B, and C, each comprising some elements drawn from U, their operations with U allow: determining what elements are present or absent across combinations, helping to map them out easily in binary representations.

In our exercise, we considered the universal set \( U = \{a, b, c, d, e, f, g, h\} \). Elements from this set are used to determine the binary occurrence of an element within subsets like A-B. Universal sets provide a structured way to manage and analyze relationships between different sets in such exercises.

Set Operation

Set operations are essential tools in mathematics, allowing us to manipulate and understand relationships between various sets. Common operations include union, intersection, and difference (or relative complement).

• Union combines all elements from given sets.
• Intersection identifies common elements between sets.
• Difference (or relative complement) discovers elements in one set but not in another.

In this exercise, the operation \( A-B \), which is a form of set difference, was used. It seeks to find elements that exist in set A but do not appear in set B. This operation can help in distinguishing unique characteristics of sets or removing shared elements to highlight differences. By mastering set operations, you can analyze and solve complex problems in set theory more effectively.