Question

Let POPULATION be the set of all people. Let \(R\) be the binary relation on \(P O P U\) LATION such that \((x, y) \in R\) if \(x\) is an older brother of \(y\) or \(x=y\). Is \(R\) refiexive? Symmetric? Antisymmetric? Transitive? An equivalence relation?

Short answer

R is reflexive, antisymmetric, and transitive, but not symmetric or an equivalence relation.

Step-by-step solution

Analyzing Reflexivity

A relation is reflexive if every element is related to itself. Here, for every person \(x\) in POPULATION, \((x, x)\) must be in \(R\). Since the condition \(x = x\) is true for all \(x\), \(R\) is reflexive because everyone is equivalent to themselves in any context.

Checking Symmetry

A relation is symmetric if for every \((x, y)\) in \(R\), \((y, x)\) is also in \(R\). In this case, if \(x\) is an older brother of \(y\) (thus \((x, y)\) is in \(R\)), \((y, x)\) may not necessarily be in \(R\) since \(y\) cannot be the older brother of \(x\). Therefore, \(R\) is not symmetric.

Evaluating Antisymmetry

A relation is antisymmetric if whenever \((x, y)\) and \((y, x)\) are in \(R\), then \(x = y\). If \((x, y)\) indicates \(x\) is an older brother of \(y\), \((y, x)\) would imply \(y\) is an older brother of \(x\). But this is not possible unless \(x = y\). Thus, \(R\) is antisymmetric.

Examining Transitivity

A relation is transitive if whenever \((x, y)\) and \((y, z)\) are in \(R\), then \((x, z)\) is also in \(R\). If \(x\) is an older brother of \(y\) and \(y\) is an older brother of \(z\), \(x\) should be an older brother of \(z\). This holds true, making \(R\) transitive.

Determining if R is an Equivalence Relation

An equivalence relation must be reflexive, symmetric, and transitive. Since \(R\) is not symmetric, it does not fulfill the requirements of an equivalence relation.

Key concepts

Reflexive Relation

A reflexive relation is one where every element in the set is related to itself.
In the context of the given problem with the set of all people, POPULATION, a binary relation \( R \) is defined.
For \( R \) to be reflexive, each person \( x \) in the POPULATION should satisfy the condition \( (x, x) \in R \).

Within our problem, this means that everyone is considered to be their own equivalent when you look at the relation defined: being one's own older brother doesn't make sense, but the identity relation does.
Essentially, it is confirmed by the fact that \( x = x \) for all \( x \), satisfying the reflexive condition.
Reflexivity is a fundamental property because it assigns the relation to self-correspondence.
It's like saying each person recognizes themselves when they see their own reflection!

Antisymmetric Relation

An antisymmetric relation has specific criteria: for a relation \( R \), if both \( (x, y) \) and \( (y, x) \) are in \( R \), then \( x \) must equal \( y \).
Let's break it down in simpler terms with our scenario. If person \( x \) is an older brother to another person \( y \), consider the reverse: \( y \) being an older brother to \( x \) would contradict, considering the definition of 'older'.

Practically, if \( x \) is the older brother in our given relation \((x, y) \), and there's no possibility for \( y \) to be an older brother according to \((y, x) \) unless \( x = y \).
Hence, the condition of antisymmetry is satisfied.
In this specific relation, it neatly outlines an order where one cannot reverse roles unless they are the same individual. This explains why antisymmetry holds in brotherly order relations and underscores why not all relations are interchangeable in direction.

Transitive Relation

A relation is transitive if the condition extends through connections: if \( (x, y) \) and \( (y, z) \) are in \( R \), then \( (x, z) \) should also be in \( R \).
Let's explore this in the context provided. Imagine \( x \) is an older brother of \( y \), and then \( y \) is an older brother of \( z \).
This setup implies that \( x \) must then be an older brother of \( z \) to maintain order through transitivity.

This makes perfect sense in family hierarchies, where the older status carries over through intermediate siblings.
Therefore, the condition that \( x \) is an older brother to \( z \) if \( (x, y) \) and \( (y, z) \) hold, confirms \( R \) as a transitive relation.
It's like a chain of command, where the oldest sibling influences across multiple steps. This aspect of order helps ensure consistency when tracking relationships across several links.