Q.The ordered pair $(5, 2)$ belongs to the relation $R = \{(x, y) : y = x - 5,\ x, y \in \mathbf{Z}\}$

The relation $R = \{(x, y) : y = x - 5,\ x, y \in \mathbf{Z}\}$ consists of all ordered pairs of integers where the second coordinate equals the first coordinate minus 5.

Check the condition:

For the pair $(5, 2)$ to belong to $R$, it must satisfy $y = x - 5$ with $x = 5$ and $y = 2$.

Substitute: $2 = 5 - 5 = 0$.

This gives $2 = 0$, which is false. …

Understanding Relations and Membership

A relation $R$ on the integers is simply a set of ordered pairs. The notation $R = \{(x, y) : y = x - 5,\ x, y \in \mathbf{Z}\}$ tells us that $R$ consists of all ordered pairs $(x, y)$ where both coordinates are integers and the second coordinate equals the first coordinate minus $5$.

To decide whether a specific pair belongs to $R$, we need to verify two things:

• Both components are integers (the domain requirement)
• The pair satisfies the defining equation $y = x - 5$

Think of the relation as a rule: "Take any integer, subtract $5$, and pair the original with the result." So $(7, 2)$ would be in $R$ because $2 = 7 - 5$. Similarly, $(0, -5)$ is in $R$ because $-5 = 0 - 5$. The question asks us to test whether $(5, 2)$ passes this rule.

Verification

1. Check the domain requirement

Both $5$ and $2$ are integers, so the pair $(5, 2)$ is at least a candidate for membership in $R$. We can proceed to the defining condition.

2. Test the relation's defining equation

For $(5, 2)$ to belong to $R$, we need $y = x - 5$ to hold when $x = 5$ and $y = 2$.

Substituting:

$2 \stackrel{?}{=} 0$ …