Problem 4 - Applied Discrete Structures

Question

Let \(A\) be the set of strings of \(0^{\prime} s\) and \(1^{\prime} s\) of length 3 or less. (a) Define the relation of \(d\) on \(A\) by \(x d y\) if \(x\) is contained within \(y\). For example, 01d101. Draw a digraph for this relation. (b) Do the same for the relation \(p\) defined by \(x p y\) if \(x\) is a prefix of \(y\). For example, \(10 p 101\), but \(01 p 101\) is false.

Step-by-Step Solution

Answer

(a) Draw edges for substring relations; (b) Draw edges for prefix relations.

Step 1: Enumerate the Elements of Set A

The set \( A \) consists of binary strings with lengths from 0 up to 3. Therefore, the elements of \( A \) are \( \{\varepsilon, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111\} \), where \( \varepsilon \) represents the empty string.

Step 2: Define Relation d (Substring)

The relation \( d \) is defined such that \( x d y \) if \( x \) is a substring of \( y \). This means for every element \( x \) in \( A \), you should identify which strings in \( A \) contain \( x \) as a substring.

Step 3: Draw the Digraph for Relation d

Each element of \( A \) will have a vertex. Draw a directed edge \( x \to y \) if \( x d y \). For example, draw edges: \( 0 \to 00, 0 \to 01, 01 \to 101\), etc. Continue this for all string pairs where \( x \) is a substring of \( y \).

Step 4: Define Relation p (Prefix)

The relation \( p \) is defined such that \( x p y \) if \( x \) is a prefix of \( y \). This means for every element \( x \) in \( A \), determine which strings \( y \) start with \( x \).

Step 5: Draw the Digraph for Relation p

Draw a vertex for each string. Draw a directed edge \( x \to y \) if \( x \) is a prefix of \( y \). For instance, draw edges: \( 1 \to 10, 1 \to 11, 10 \to 101\), etc. Do this for all valid pairs \( (x, y) \) where \( x \) is a prefix of \( y \).

Key Concepts

Digraph

A digraph, or directed graph, is a graphical representation of binary relations between elements of a set. In the context of our exercise, each element of the set \( A \)—consisting of binary strings—is represented as a vertex in the digraph. Here's how it works:

• A node for every member in your set \( A \).
• A directed edge \( x \rightarrow y \) signifies that 'x' has a relation to 'y'.

The direction is important because it shows the flow of the relation.Take, for example, the relation \( d \), where if \( x d y \), then \( x \) is a substring of \( y \). This means you would draw an edge from the node representing \( x \) to the node representing \( y \) if such a substring relationship exists.Therefore, a digraph visually tracks how different elements in a set relate to each other through defined relations like 'substring' or 'prefix'. It is a powerful tool to quickly see connections and understand the relationships at play.

Substring

A substring is a sequence of characters that appears in the same order within another string. This is crucial in defining the relation \( d \) for the set \( A \). For instance:

• If you have the string \( 010 \), substrings include \( 0, 1, 01, 10, \) and \( 010 \).
• Every string is a substring of itself.

In set \( A \), defined as all binary strings of length up to 3, each element can be checked to see if it appears in a larger string also within set \( A \). This relationship allows us to map out how various strings interrelate by appearance.For example, in this context, \( 01 \) is a substring of \( 101 \), so the edge \( 01 \rightarrow 101 \) will appear in the digraph for relation \( d \). Such examinations reveal inherent relations that might not be immediately obvious, providing deeper understanding of how strings can be parsed and combined.

Prefix

A prefix relates to how a string begins and is used to define the relation \( p \) within our set \( A \). A string \( x \) is a prefix of another string \( y \) if \( y \) starts with \( x \). Consider:

• The string \( 10 \) is a prefix of \( 101 \).
• Every string is a prefix of itself.
• An empty string \( \varepsilon \) is a prefix to any string.

Use these rules when determining the relationships for drawing a digraph representing the prefix relation \( p \). For example, if \( x \) is a prefix of \( y \), an edge is drawn from \( x \) to \( y \) in this new digraph.By examining prefixes, one can uncover how strings are layered, witnessing the incremental building of a string starting from its prefix. This approach is pivotal for coding and language processing, where understanding and extracting prefixes can benefit applications significantly.

Set Theory

Set theory is the backbone of dealing with collections of items—known as sets—and understanding their properties and relationships. In the exercise, set \( A \) is defined as set of all strings of 0s and 1s with length up to 3. Let's break it down:

• Elements in a set do not repeat. \( \{0, 0, 1, 1\} \) would be \( \{0, 1\} \).
• Set operations like union, intersection, and difference can be performed.
• Sets can be finite or infinite based on the number of elements.

Using set theory rules, one can confidently define relations on sets—such as the 'substring' or 'prefix' relations used in this exercise. It provides a universal language to express mathematical concepts and ideas formally. With concepts such as subsets and power sets, exploring relations within set \( A \) becomes systematic, making visual representations like digraphs intuitive to draw and interpret.