Understanding Relations on a Set

The question asks us to analyze a specific relation defined on a set X, which represents all persons living in a city. The relation, denoted as $x < y$, holds if person $y$ is at least 5 years older than person $x$. We need to determine if this relation is reflexive, symmetric, or transitive.

Let's denote the age of a person $p$ as $\text{age}(p)$. The relation $x < y$ means $\text{age}(y) \ge \text{age}(x) + 5$. We will examine each property of a relation.

Checking for Reflexivity of the Relation

A relation R on a set X is called reflexive if for every element $x \in X$, the relation $x \text{ R } x$ holds. In our case, this means for every person $x$ in the city, the relation $x < x$ must be true.

According to the definition of our relation, $x < x$ would mean that person $x$ is at least 5 years older than person $x$. This translates to $\text{age}(x) \ge \text{age}(x) + 5$.

Let's test this inequality:

$\text{age}(x) \ge \text{age}(x) + 5$

Subtracting $\text{age}(x)$ from both sides gives:

$0 \ge 5$

This inequality $0 \ge 5$ is clearly false. A person cannot be 5 years older than themselves. Therefore, the relation $x < y$ (where $y$ is at least 5 years older than $x$) is not reflexive.

Checking for Symmetry of the Relation

A relation R on a set X is called symmetric if for every $x, y \in X$, whenever $x \text{ R } y$ holds, it implies that $y \text{ R } x$ also holds. For our relation $x < y$, this means if $x < y$ is true, then $y < x$ must also be true.

Let's assume $x < y$ is true. By the definition, this means person $y$ is at least 5 years older than person $x$, i.e., $\text{age}(y) \ge \text{age}(x) + 5$.

Now, for the relation to be symmetric, $y < x$ must also be true. By definition, $y < x$ means person $x$ is at least 5 years older than person $y$, i.e., $\text{age}(x) \ge \text{age}(y) + 5$.

Let's consider an example. Suppose person A is 30 years old and person B is 37 years old. Here, $\text{age}(B) = 37$ and $\text{age}(A) = 30$.

• Is A < B true? $\text{age}(B) \ge \text{age}(A) + 5$? $37 \ge 30 + 5$? $37 \ge 35$? Yes, this is true. So, A < B holds.
• Is B < A true? $\text{age}(A) \ge \text{age}(B) + 5$? $30 \ge 37 + 5$? $30 \ge 42$? No, this is false. So, B < A does not hold.

Since we found a case where A < B holds but B < A does not hold, the relation is not symmetric.

Checking for Transitivity of the Relation

A relation R on a set X is called transitive if for every $x, y, z \in X$, whenever $x \text{ R } y$ and $y \text{ R } z$ hold, it implies that $x \text{ R } z$ also holds. For our relation $x < y$, this means if $x < y$ and $y < z$ are true, then $x < z$ must also be true.

Let's assume $x < y$ and $y < z$ are both true.

From $x < y$, we have $\text{age}(y) \ge \text{age}(x) + 5$.

From $y < z$, we have $\text{age}(z) \ge \text{age}(y) + 5$.

We want to check if $x < z$ is true, which means $\text{age}(z) \ge \text{age}(x) + 5$.

Let's use the two inequalities:

1) $\text{age}(y) \ge \text{age}(x) + 5$

2) $\text{age}(z) \ge \text{age}(y) + 5$

Substitute the first inequality into the second one (replace $\text{age}(y)$ with its minimum value from inequality 1):

$\text{age}(z) \ge (\text{age}(x) + 5) + 5$

$\text{age}(z) \ge \text{age}(x) + 10$

If $\text{age}(z)$ is greater than or equal to $\text{age}(x) + 10$, it must also be greater than or equal to $\text{age}(x) + 5$. That is, if $\text{age}(z) \ge \text{age}(x) + 10$, then $\text{age}(z) \ge \text{age}(x) + 5$ is automatically true.

So, if $x < y$ and $y < z$ hold, it implies $x < z$ holds. Therefore, the relation is transitive.

Conclusion on Relation Properties

Based on our analysis of the relation $x < y$ where $y$ is at least 5 years older than $x$ on the set of persons X:

• The relation is not reflexive because a person is not at least 5 years older than themselves.
• The relation is not symmetric because if $y$ is at least 5 years older than $x$, $x$ cannot be at least 5 years older than $y$.
• The relation is transitive because if $y$ is at least 5 years older than $x$, and $z$ is at least 5 years older than $y$, then $z$ must be at least 5 years older than $x$ (in fact, at least 10 years older).

Therefore, the correct description of the relation is that it is transitive but neither reflexive nor symmetric. This matches the description in one of the provided options.

Relation Properties Revision Table

Additional Information on Relation Types

Understanding different types of relations is fundamental in discrete mathematics and set theory. Here are some common types:

• Equivalence Relation: A relation that is simultaneously reflexive, symmetric, and transitive. Such a relation partitions the set into disjoint equivalence classes. The given relation is not an equivalence relation because it is neither reflexive nor symmetric.
• Partial Order Relation: A relation that is reflexive, antisymmetric, and transitive. Antisymmetric means if $x \text{ R } y$ and $y \text{ R } x$ both hold, then $x=y$. The given relation is not antisymmetric (consider $x=y$, $\text{age}(y) \ge \text{age}(x)+5$ is false). The relation "is older than or same age as" ($\text{age}(y) \ge \text{age}(x)$) is a partial order.
• Strict Partial Order Relation: A relation that is irreflexive (nothing is related to itself) and transitive. Asymmetric relations are also irreflexive. An asymmetric relation is one where if $x \text{ R } y$ holds, $y \text{ R } x$ cannot hold. Our relation is not strictly asymmetric because if $\text{age}(y) = \text{age}(x)+5$, A < B holds, but B < A does not. If we used "strictly older than" (age(y) > age(x)), it would be asymmetric and thus irreflexive. With "at least 5 years older," it is irreflexive but not strictly asymmetric. However, it is transitive.

The relation "is at least 5 years older than" falls under the category of relations that are transitive but lack reflexivity and symmetry.