Question

Show that composition of relations is an associative operation. That is, show that if \(R, S,\) and \(T\) are binary relations on a set \(X,\) then $$R \circ(S \circ T)=(R \circ S) \circ T$$

Short answer

Composition of relations is associative.

Step-by-step solution

Understand the Definitions

Given binary relations \(R, S,\) and \(T\) on a set \(X\), we need to show that the composition \((R \circ S) \circ T\) is equal to \(R \circ (S \circ T)\). The composition of relations is defined as for \(x, y \in X\), \(x (R \circ S) y\) holds if there exists a \(z \in X\) such that \(x R z\) and \(z S y\). Similarly, \(x (S \circ T) y\) holds if there exists a \(w \in X\) such that \(x S w\) and \(w T y\).

Express \((R \circ S) \circ T\)

For \((R \circ S) \circ T\), this means: there exists some \(a \in X\) such that \(x (R \circ S) a\) and \(a T y\). For \(x (R \circ S) a\), there must exist some \(b \in X\) such that \(x R b\) and \(b S a\). Thus, for \((R \circ S) \circ T\), \(x R b\), \(b S a\), and \(a T y\) must hold for some \(b, a \in X\).

Express \(R \circ (S \circ T)\)

For \(R \circ (S \circ T)\), this means: there exists some \(c \in X\) such that \(x R c\) and \(c (S \circ T) y\). For \(c (S \circ T) y\), there must exist some \(d \in X\) such that \(c S d\) and \(d T y\). Thus, for \(R \circ (S \circ T)\), \(x R c\), \(c S d\), and \(d T y\) must hold for some \(c, d \in X\).

Demonstrate Equivalence

Notice in both descriptions, the main condition is the existence of intermediate elements from \(X\) that make the sequence of relations hold. Particularly, for \((R \circ S) \circ T\): we need \(b, a \) where \(x R b, b S a, a T y\) and for \(R \circ (S \circ T)\): we seek \(c, d\) where \(x R c, c S d, d T y\). The chains of relations can be adjusted such that \(b = c\) and \(a = d\). This illustrates that both compositions are fundamentally making the same relational connections in \(X\).

Conclude Associativity

Since we have shown that the elements connecting through \((R \circ S) \circ T\) can be structurally mapped to those of \(R \circ (S \circ T)\), we conclude \(x((R \circ S) \circ T)y\) iff \(x(R \circ (S \circ T))y\) for all \(x, y \in X\). Thus, \(R \circ (S \circ T) = (R \circ S) \circ T\), proving the associativity of relation composition.

Key concepts

Composition of Relations

The composition of relations is a crucial operation in set theory. It involves combining two relations to form a new one. You can think of it as a way to "chain" relations together over elements of a set. For two relations, say \( R \) and \( S \) on a set \( X \), the composition \((R \circ S)\) means that for any elements \( x, y \) in the set, there must be an intermediate element \( z \) such that \( x R z \) and \( z S y \) hold true.
This shows a path from \( x \) to \( y \) through \( z \). This chaining effect is pivotal when you have multiple relations. It allows going from one element to another indirectly.
When we talk about associativity in compositions, it means that when you have multiple compositions, like \((R \circ S) \circ T\) and \(R \circ (S \circ T)\), the order in which you perform them doesn't matter. They will both result in the same relation. This is what associativity of composition of relations guarantees.

Binary Relations

Binary relations are a fundamental concept that links pairs of elements from a set. For a set \( X \), a binary relation \( R \) is a subset of the Cartesian product \( X \times X \). This means that if \( R \) is a relation, it consists of ordered pairs \( (x, y) \) where \( x \) and \( y \) are in \( X \) and there is a connection from \( x \) to \( y \).
Binary relations are everywhere in mathematics and computer science. They help to model relational databases, graphs, and various logical expressions.
When you deal with sets in set theory, understanding how binary relations can connect elements offers a deeper grasp on complex concepts such as orderings, equivalence relations, and functions. These relations can be reflexive, symmetric, or transitive, contributing to the structure and properties of the set.

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. It serves as a foundation for most mathematical disciplines. Sets can be anything from numbers to symbols to more abstract concepts.
In set theory, you explore relations, functions, and operations among sets. Consider a set \( X \); relations among its elements reveal interconnectedness. Relations, particularly binary ones, are represented as certain subsets of \( X \times X \).
Operations on sets include union, intersection, and difference, whereas relations, such as composition, also provide powerful ways to combine and manipulate these sets. Set theory provides a systematic way to navigate and organize information by understanding these connections, helping to solve problems in areas like logic, statistics, and computer science.