Problem 25 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The characteristic function h of the set \(\{b, c, g\}\) is given by: \[h(x) = \begin{cases} 1, & \text{if}\ x = b, c, g \\ 0, & \text{otherwise} \end{cases}\]

Step 1: Write down the given sets

The universal set U is given as: \[U = \{a, b, c, d, e, f, g\}\] And the set for which we need to find the characteristic function is: \[A = \{b, c, g\}\]

Step 2: Define the characteristic function h

Now we'll define the characteristic function h of the set A. The characteristic function h maps elements from the universal set U to the set {0, 1}: \[h(x) = \begin{cases} 1, & \text{if}\ x \in A \\ 0, & \text{if}\ x \notin A \end{cases}\]

Step 3: Find the values of the characteristic function for each element in set U

Next, we'll find the values of the function h for each element in the universal set U: \(h(a) = 0\) since a is not in set A. \(h(b) = 1\) since b is in set A. \(h(c) = 1\) since c is in set A. \(h(d) = 0\) since d is not in set A. \(h(e) = 0\) since e is not in set A. \(h(f) = 0\) since f is not in set A. \(h(g) = 1\) since g is in set A.

Step 4: Write down the characteristic function of set A

Now, we can write the characteristic function h of the set A as: \[h(x) = \begin{cases} 1, & \text{if}\ x = b, c, g \\ 0, & \text{otherwise} \end{cases}\] Thus, the characteristic function h of the set {b, c, g} has been defined.

Key Concepts

Universal Set

The universal set is an essential concept in set theory. Specifically, it is the set that contains all the objects or elements under consideration for a particular discussion or problem. In our exercise, the universal set, denoted by \( U \), contains the elements \( \{a, b, c, d, e, f, g\} \). This means that any subset we discuss, including the set \( A = \{b, c, g\} \), is drawn from \( U \).
Understanding the universal set is crucial as it defines the total "universe" we are working within. Whether you are dealing with number theory, geometry, or any kind of real-world data, there's always a universal set that frames the elements you're discussing.

• The universal set includes everything needed for a particular problem.
• It helps define the scope and limits of any subset.

In practical applications, choosing the correct universal set ensures clarity and limits misunderstandings in mathematical discussions.

Set Theory

Set theory is the branch of mathematics that studies collections of objects, which are called sets. In the simplest terms, a set is just a collection of distinct items, typically called elements or members. Set theory allows us to group objects together, often according to certain rules or characteristics. In our problem, the sets \( U \) and \( A \) are used to work through the solution.
Some basic operations within set theory include:

• Union: Combines all elements from both sets into a new set.
• Intersection: Finds common elements shared between sets.
• Difference: Elements present in one set but not in another.

In our specific context, we relied on understanding the given sets and how they are related. The characteristic function, which maps elements from the universal set \( U \) to \( \{0, 1\} \), is a good example of set theory's application. By defining and analyzing sets, mathematicians can solve problems involving unknown groups of data, whether they relate to numbers, colors, shapes, or real-world concepts.

Function Mapping

Function mapping is particularly about how each element of one set is paired to an element of another set. In our scenario, we used a specific type of function mapping known as a characteristic function. It is fundamental in set theory. The characteristic function \( h \) maps elements from the universal set \( U \) to the set \( \{0, 1\} \), providing a way to specify membership in a set.
Here is how the characteristic function works:

• If an element \( x \) belongs to the subset \( A \), \( h(x) = 1 \).
• If \( x \) does not belong to \( A \), \( h(x) = 0 \).

This mapping function visually and mathematically represents which elements belong to a subset \( A \) from the universal set \( U \).
Function mapping plays a critical role not just in set theory but across various mathematical disciplines, including calculus and analysis, because it provides a framework for transforming and translating sets and their elements into different contexts or dimensions.