Problem 23 - Discrete Mathematics with Applications

Question

Let \(U=\\{\mathrm{a}, \ldots, \mathrm{g}\\} .\) Define the characteristic function \(h\) of each set. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}, \mathrm{f}\\}$$

Step-by-Step Solution

Answer

The characteristic function \(h(x)\) for the set \(\{a, c, d, f\}\) within the universal set \(U = \{a, \ldots, g\}\) is defined as: \(h(x) = \begin{cases} 1, & \text{if}\ x \in \{a, c, d, f\} \\ 0, & \text{otherwise} \end{cases}\)

Step 1: Understand the Universal Set U

The given universal set U consists of elements a, b, c, d, e, f, and g. All other sets in this exercise will be subsets of U.

Step 2: Define the Given Set

The set we need to find the characteristic function for is given as {a, c, d, f}. This set is a subset of U which includes elements a, c, d, and f only.

Step 3: Define the Characteristic Function h

To define the characteristic function h for the given set, we will write a function that assigns 1 to an element if it is in the set {a, c, d, f} and 0 otherwise. Mathematically, this can be expressed as: \(h(x) = \begin{cases} 1, & \text{if}\ x \in \{a, c, d, f\} \\ 0, & \text{otherwise} \end{cases}\)

Step 4: Describe the Function h for Each Element of U

Now, we can describe the function h for each element in the universal set U: - For element a: \(h(a) = 1\) since a is in the set {a, c, d, f}. - For element b: \(h(b) = 0\) since b is not in the set {a, c, d, f}. - For element c: \(h(c) = 1\) since c is in the set {a, c, d, f}. - For element d: \(h(d) = 1\) since d is in the set {a, c, d, f}. - For element e: \(h(e) = 0\) since e is not in the set {a, c, d, f}. - For element f: \(h(f) = 1\) since f is in the set {a, c, d, f}. - For element g: \(h(g) = 0\) since g is not in the set {a, c, d, f}. So the characteristic function h for the given set {a, c, d, f} within the universal set U is: \(h(x) = \begin{cases} 1, & \text{if}\ x \in \{a, c, d, f\} \\ 0, & \text{otherwise} \end{cases}\)

Key Concepts

Universal Set

In set theory, the concept of a universal set is fundamental. It is designated as the set that includes all objects under consideration for a particular discussion or problem. Think of it as the 'whole' within which every other set exists as a 'part' or 'portion'. If we are talking about letters of the alphabet, then the universal set could be all the letters from A to Z.

For example, let's take the universal set to be all lowercase letters from a to g, represented by a set notation as follows:
\[ U = \{ a, b, c, d, e, f, g \} \].
Essentially, the universal set forms the domain for any characteristic functions we might define on subsets of that universal set. Universal sets can vary depending on the subject we are speaking about but aim to cover the 'universe' of that context.

Subset

A subset is a set every element of which is also an element of a larger set, known as the 'superset'. For instance, if we take our universal set of lowercase letters up to g, a subset would be any collection of these letters, no matter the number or order, as long as all of them are also found in the universal set.

For the set used in our exercise:
\[ \{a, c, d, f\} \]
this is a subset of \( U \) since every element in this set is also included in the universal set. One critical point to remember is that under set theory, every set is considered to be a subset of itself and the null set, which contains no elements, is also a subset of all sets, including the universal set.

Characteristic Function

A characteristic function is a mathematical way to express inclusion or exclusion of elements from a set. It is effectively a function that maps every element of the universal set to either 1 or 0. This dichotomy represents whether an element is a member of a subset (1) or not a member (0).

In our problem, the characteristic function \( h(x) \) defined for the subset \( \{a, c, d, f\} \) within the universal set \( U \) would look like this:
\[ h(x) = \begin{cases} 1, & \text{if } x \in \{a, c, d, f\} \ 0, & \text{otherwise} \end{cases} \].
The practical significance of this function is to easily identify the subset a particular element belongs to, relative to the universal set.

Set Theory

Set Theory is the branch of mathematical logic that studies collections of objects, which we refer to as sets. It deals with understanding how sets interact, the nature of their elements, and the relations between them. Foundational concepts in set theory include notions of union, intersection, difference, and complement.

Characteristic functions are part of the fabric of set theory because they provide a simple, binary way to describe sets and how they relate to each other. Through set theory, we can better understand the organization, partition, and structure of data or systems, which has implications not only for mathematics but also for computer science, probability, and more. It's a versatile tool that allows us to comprehend the fundamental nature of collections and quantities.