Problem 29 - Discrete Mathematics with Applications

Question

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{h}\\} .\) Characteristic function \(f_{S}\) is given as an 8 -bit word. Find the corresponding set \(S .\) $$10101010$$

Step-by-Step Solution

Answer

The corresponding set \(S\) based on the 8-bit word (characteristic function) \(10101010\) is $$S = \{a, c, e, g\}$$.

Step 1: Understand the representation of the characteristic function

The 8-bit word represents the presence (1) or absence (0) of elements in the set \(S\) with respect to the order of elements in the universal set \(U\).

Step 2: List all the elements of the universal set and their binary representation

Let us list all the elements of the universal set alongside their binary representation: \(a \rightarrow 1\) \(b \rightarrow 0\) \(c \rightarrow 1\) \(d \rightarrow 0\) \(e \rightarrow 1\) \(f \rightarrow 0\) \(g \rightarrow 1\) \(h \rightarrow 0\)

Step 3: Create the subset S based on the binary representation

Now, we will create a subset \(S\) of \(U\) containing elements that have a binary representation of 1 in the given characteristic function. Based on the binary representation given above, we can create the subset S: $$S = \{a, c, e, g\}$$

Key Concepts

Set Theory

Set theory is a fundamental branch of mathematics that deals with the collection of objects, which are known as elements or members of the set. These objects can be anything: numbers, symbols, points in geometry, or even other sets. The importance of set theory lies in its ability to provide a unified framework for understanding and describing all mathematical concepts.

When working with sets, certain operations such as union, intersection, and complement are essential to understand. For instance, the union of two sets contains all the elements that are in either set, while the intersection contains only those elements that are in both sets. The complement of a set includes all the elements that are not in the set within the context of a given universal set, which brings us to the concept of the characteristic function.

The characteristic function, in the context of set theory, is a way to represent a subset within a universal set using a binary format. The binary digits (1's and 0's) indicate the presence or absence of each element in the subset, which is exactly how we used it in the problem and solution presented.

Binary Representation

Binary representation is a system that allows expressing any number or code using only two symbols: 0 and 1. This binary system is of paramount importance in the realm of computers and digital electronics because it aligns with the two states of digital technology which are 'on' and 'off.'

In the given exercise, binary representation succinctly expresses which elements are included in the set by using the 8-bit sequence. The bit position corresponds to an element's position in the universal set, where a '1' means the element is included in the subset, and a '0' means it is not. By translating the binary sequence '10101010', we can infer which elements of the universal set are present in set S.

This binary approach simplifies the visual representation of sets, especially when dealing with larger universal sets or when operations between multiple sets are needed. The clear-cut nature of binary makes it easier to perform set operations like intersections, unions, and complements through bit manipulation.

Universal Set

In set theory, the universal set is a set that contains all the objects under consideration and typically defines the universe of discourse. All other sets within a problem are subsets of this universal set. It is often denoted by the symbol 'U' and helps to contextualize the discussion by establishing a scope within which everything else is defined.

The concept of a universal set is vital when discussing complements or creating characteristic functions, as it gives us the foundational pool of elements against which all other sets are measured. In the exercise, the universal set U was defined as containing the elements from 'a' to 'h.' The characteristic function was then used to identify which elements belong to the set S by associating each bit with an element in the prescribed order of U.

Understanding the role of the universal set is crucial in problems like these, as it serves as the reference point for other sets and operations, which can include unions, intersections, complements, or, as in our case, applying a characteristic function to identify specific elements.