Question

Let \(A=\\{a, b, c, d\\} .\) Define the relations \(R_{1}\) and \(R_{2}\) on \(A\) as $$R_{1}=\\{(a, a),(a, b),(b, d)\\}$$ and $$R_{2}=\\{(a, d),(b, c),(b, d),(c, b)\\}$$ Find (a) \(R_{1} \circ R_{2}\) (b) \(R_{2} \circ R_{1}\) (c) \(R_{1}^{2}\) (d) \(R_{2}^{2}\)

Short answer

(a) \(\{(a, c), (a, d)\}\); (b) \(\{(c, d)\}\); (c) \(\{(a, a), (a, d)\}\); (d) \(\{(b, b), (c, c), (c, d)\}\).

Step-by-step solution

Understand Composition of Relations

The composition of two relations \(R_1\) and \(R_2\), denoted \(R_1 \circ R_2\), is the set of ordered pairs \((x, z)\) such that there exists an element \(y\) where \((x, y) \in R_1\) and \((y, z) \in R_2\). The order is crucial since it involves checking pairs from \(R_2\) first.

Find \(R_1 \circ R_2\)

To find \(R_1 \circ R_2\), consider each pair in \(R_1\):- From \( (a,a) \) and \(R_2\), no pair leads to a new connection because \((a, a)\) does not initiate any paths.- From \((a, b)\), look for pairs in \(R_2\) starting with \(b\). There are two: \((b, c)\) and \((b, d)\). This results in pairs \((a, c)\) and \((a, d)\).- From \((b, d)\), there is no pair in \(R_2\) starting with \(d\). Thus, \(R_1 \circ R_2 = \{(a, c), (a, d)\}\).

Find \(R_2 \circ R_1\)

To find \(R_2 \circ R_1\), consider each pair in \(R_2\):- From \((a, d)\), there's no starting pair in \(R_1\) with \(d\). - From \((b, c)\), there's no \( (c, x) \) in \(R_1\).- From \((b, d)\), there's no \((d, x)\) in \(R_1\).- From \((c, b)\), you find \((b, d)\) in \(R_1\), leading to \((c, d)\).Thus, \(R_2 \circ R_1 = \{(c, d)\}\).

Find \(R_1^2\)

The square of a relation \(R_1^2\) is \(R_1 \circ R_1\). Evaluating:- From \((a, a)\), can directly go \((a, a)\).- From \((a, b)\), follow up with \((b, d)\) results in \((a, d)\).- From \((b, d)\), there is no \((d, x)\) in \(R_1\).Thus, \(R_1^2 = \{(a, a), (a, d)\}\).

Find \(R_2^2\)

The square of \(R_2^2\) is \(R_2 \circ R_2\). Evaluating:- From \((a, d)\), no \((d, x)\) starts in \(R_2\).- From \((b, c)\) followed by \((c, b)\) results in \((b, b)\).- From \((b, d)\), no \((d, x)\) starts in \(R_2\).- From \((c, b)\), find \((b, c)\) leading to \((c, c)\), also find \((b, d)\) leading to \((c, d)\).Thus, \(R_2^2 = \{(b, b), (c, c), (c, d)\}\).

Key concepts

Relations

In discrete mathematics, a relation is a key concept that defines a way to associate elements from one set with elements of another. A relation from set \( A \) to set \( B \) is a subset of the Cartesian product \( A \times B \). This means it pairs elements from \( A \) with elements from \( B \), expressed as ordered pairs.
The relation itself is the collection of these ordered pairs, and it usually depends on some rule or condition that determines which pairs are included. For example, given two sets \( A \) and \( B \), we say an element \( a \) in \( A \) is related to an element \( b \) in \( B \) if the pair \((a, b)\) belongs to the relation.
Relations have properties such as reflexivity, symmetry, and transitivity that help determine the type of relation. These properties can significantly aid in understanding how elements are related within a set or between multiple sets.

Composition of Relations

The composition of relations is a process used to combine two relations to form a new one. Denoted as \( R_1 \circ R_2 \), this composition helps determine how pairs can be connected through an intermediate element. When composing two relations, you look for an intermediary element that creates a new connection.
Here's how it works: if \( (x, y) \) is an element of relation \( R_1 \) and \( (y, z) \) is an element of relation \( R_2 \), then \( (x, z) \) becomes an element of the composed relation. It's like finding a path from \( x \) to \( y \) and from \( y \) to \( z \) and summarizing it as a direct path from \( x \) to \( z \).

• The composition is not necessarily commutative, meaning \( R_1 \circ R_2 \) is not always the same as \( R_2 \circ R_1 \).
• This property allows different sequences of relationships to lead to potentially different results.

Understanding the composition of relations is essential in fields such as database management, graph theory, and more.

Set Theory

Set theory, a fundamental part of discrete mathematics, deals with the study and understanding of sets. Sets are collections of distinct objects or elements considered as a single entity.
The theory introduces notations and operations like union, intersection, and difference, which are used to analyze and manipulate these collections. Relations, as discussed earlier, are closely tied to set theory as they are expressed using ordered pairs, which are themselves subsets of Cartesian products of sets.

• In set theory, elements are often denoted by lowercase letters, whereas sets are represented by uppercase letters.
• Set operations provide the tools to systematically combine, relate, and compare sets, forming the foundation for more complex mathematical structures.

Understanding set theory is crucial for anyone delving into mathematics or computer science, as it forms the basis for logic, probability, and many other fields.

Ordered Pairs

An ordered pair is a fundamental concept in set theory and relations. It's a pair of elements where the order of the elements is crucial. That is, \((a, b)\) is not necessarily the same as \((b, a)\).
Ordered pairs allow us to establish relationships between two elements. They form the backbone of defining relations and functions within sets. When dealing with complex relations, the arrangement of elements provides necessary information about how each is related.

• The first element of an ordered pair is often referred to as the "first component" or "abscissa," while the second element is the "second component" or "ordinate."
• These pairs form the basis for coordinates in geometry and are used in defining mappings or relationships in various mathematical contexts.
• Applications include graphing data, defining mathematical objects like vectors, and representing relationships in logic and computer science.

The significance of ordered pairs extends beyond just mathematics, influencing fields such as economics and physics.