Problem 4 - Applied Discrete Structures

Question

Let \(A\) be a set and let \(S\) be any subset of \(A\). Let \(\chi_{S}: A \rightarrow\\{0,1\\}\) be defined by $$ \chi_{S}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \in S \\ 0 & \text { if } x \notin S \end{array}\right. $$ The function \(\chi_{S}\) is called the characteristic function of \(S\). (a) If \(A=\\{a, b, c\\}\) and \(S=\\{a, b\\}\), list the elements of \(\chi_{S}\). (b) If \(A=\\{a, b, c, d, e\\}\) and \(S=\\{a, c, e\\},\) list the elements of \(\chi_{S}\). (c) If \(A=\\{a, b, c\\},\) what are \(\chi_{\emptyset}\) and \(\chi_{A} ?\)

Step-by-Step Solution

Answer

(a) \((a,1), (b,1), (c,0)\); (b) \((a,1), (b,0), (c,1), (d,0), (e,1)\); (c) \(\chi_{\emptyset} = \{(a,0), (b,0), (c,0)\}\), \(\chi_{A} = \{(a,1), (b,1), (c,1)\}\).

Step 1: Understanding the Characteristic Function

The characteristic function \( \chi_{S} \) is designed to indicate membership within a set \( S \). For any element \( x \) in the universal set \( A \), \( \chi_{S}(x) = 1 \) if \( x \in S \) and \( \chi_{S}(x) = 0 \) if \( x otin S \). This framework will be applied to analyze the specified problems for given sets.

Step 2: Solving Part (a)

Given \( A = \{a, b, c\} \) and \( S = \{a, b\} \), for each element in \( A \):- \( \chi_{S}(a) = 1 \) because \( a \in S \).- \( \chi_{S}(b) = 1 \) because \( b \in S \).- \( \chi_{S}(c) = 0 \) because \( c otin S \).Thus, the characteristic function \( \chi_{S} \) is \( \{(a,1), (b,1), (c,0)\} \).

Step 3: Solving Part (b)

Given \( A = \{a, b, c, d, e\} \) and \( S = \{a, c, e\} \), evaluate:- \( \chi_{S}(a) = 1 \) since \( a \in S \).- \( \chi_{S}(b) = 0 \) since \( b otin S \).- \( \chi_{S}(c) = 1 \) since \( c \in S \).- \( \chi_{S}(d) = 0 \) since \( d otin S \).- \( \chi_{S}(e) = 1 \) since \( e \in S \).The characteristic function \( \chi_{S} \) is \( \{(a,1), (b,0), (c,1), (d,0), (e,1)\} \).

Step 4: Solving Part (c) for Empty Set and Set A

For \( S = \emptyset \) with \( A = \{a, b, c\} \), none of the elements are in \( S \):- \( \chi_{S}(a) = 0 \);- \( \chi_{S}(b) = 0 \);- \( \chi_{S}(c) = 0 \). So, \( \chi_{\emptyset} = \{(a,0), (b,0), (c,0)\} \).For \( S = A \), each element of \( A \) is in \( S \):- \( \chi_{S}(a) = 1 \);- \( \chi_{S}(b) = 1 \);- \( \chi_{S}(c) = 1 \). Therefore, \( \chi_{A} = \{(a,1), (b,1), (c,1)\} \).

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that deals with the concept of sets, which are essentially collections of distinct objects known as elements. Sets are a fundamental concept not only in mathematics but also in computer science, logic, and various other disciplines. Sets can range from finite, like the set of all letters in the English alphabet, to infinite, like the set of all natural numbers.

In set theory:

• The notation \(x \in A\) means that \(x\) is an element of the set \(A\).
• The empty set, denoted by \(\emptyset\), is a special set that contains no elements.

Sets can be used to model collections of objects in the real world as well as abstract concepts in mathematics. Understanding how sets work is crucial for grasping more complex mathematical structures and relationships.

Subsets

Subsets are an integral concept within set theory. A subset is a set whose elements are all contained within another set. Let's say \( S \) and \( A \) are sets. If every element of \( S \) is also an element of \( A \), then \( S \) is a subset of \( A \), denoted by \( S \subseteq A \).

Key points about subsets:

• Any set \( A \) is a subset of itself (A \subseteq A).
• The empty set is a subset of any set, including itself.

Understanding subsets is important because it allows us to explore how different sets relate to each other, particularly when examining unions, intersections, and set differences.

For example, if \( A = \{a, b, c\} \) and \( S = \{a, b\} \), \( S \) is a subset of \( A \) because all elements of \( S \) are also in \( A \).

Indicator Functions

Indicator functions, often referred to as characteristic functions, are used in mathematics to examine membership of elements in a set. Specifically, the indicator function \( \chi_{S} \) maps elements from a bigger set \( A \) to either 0 or 1, depending on whether the element belongs to a subset \( S \) of \( A \).

Characteristics:

• If an element \( x \in S \), then \( \chi_{S}(x) = 1 \).
• If an element \( x otin S \), then \( \chi_{S}(x) = 0 \).

This simple classification helps in quickly identifying which elements are part of the subset \( S \). Indicator functions are not only useful in pure mathematics but also have applications in probability and statistics, where they are used to define events and simplify complex expressions.

For instance, given \( A = \{a, b, c, d, e\} \) and a subset \( S = \{a, c, e\} \), the characteristic function \( \chi_{S} \) would assign 1 to \( a, c, e \) and 0 to \( b, d \). This representation makes it easy to visualize and work with data related to membership in subset \( S \).