Understanding the Relation Re

The question defines a relation \(Re\) on the set of real numbers \(\mathbb{R}\). The relation is given by \(Re = \{(x, y) \in \mathbb{R}^2 \mid |x - y| < \epsilon\}\), where \(\epsilon = 0.0005\). This relation means that two real numbers \(x\) and \(y\) are related if the absolute difference between them is less than a small positive value \(\epsilon\). This can be interpreted as \(x\) and \(y\) being "approximately equal". We need to determine if this relation is reflexive, symmetric, and transitive.

Checking for Reflexivity

A relation \(R\) on a set \(A\) is reflexive if for every element \(a \in A\), \((a, a) \in R\).

For the relation \(Re\), we need to check if for every real number \(x \in \mathbb{R}\), \((x, x) \in Re\). According to the definition of \(Re\), this means checking if \(|x - x| < \epsilon\).

Let's evaluate \(|x - x|\):

\(|x - x| = |0| = 0\)

We are given \(\epsilon = 0.0005\). So, we need to check if \(0 < 0.0005\).

Since \(0\) is indeed less than \(0.0005\), the condition \(|x - x| < \epsilon\) is true for all real numbers \(x\).

Therefore, the relation \(Re\) is reflexive.

Checking for Symmetry

A relation \(R\) on a set \(A\) is symmetric if for every \(a, b \in A\), whenever \((a, b) \in R\), it follows that \((b, a) \in R\).

For the relation \(Re\), we need to check if for every \(x, y \in \mathbb{R}\), whenever \((x, y) \in Re\), it follows that \((y, x) \in Re\).

Assume that \((x, y) \in Re\). By the definition of \(Re\), this means \(|x - y| < \epsilon\).

Now consider the condition for \((y, x) \in Re\), which is \(|y - x| < \epsilon\).

We know that the absolute value property \(|a - b| = |b - a|\) holds for any real numbers \(a\) and \(b\).

So, \(|y - x| = |-(x - y)| = |x - y|\).

Since we assumed \(|x - y| < \epsilon\), it directly follows that \(|y - x| < \epsilon\).

Therefore, if \((x, y) \in Re\), then \((y, x) \in Re\). The relation \(Re\) is symmetric.

Checking for Transitivity

A relation \(R\) on a set \(A\) is transitive if for every \(a, b, c \in A\), whenever \((a, b) \in R\) and \((b, c) \in R\), it follows that \((a, c) \in R\).

For the relation \(Re\), we need to check if for every \(x, y, z \in \mathbb{R}\), whenever \((x, y) \in Re\) and \((y, z) \in Re\), it follows that \((x, z) \in Re\).

Assume that \((x, y) \in Re\) and \((y, z) \in Re\).

By the definition of \(Re\), this means:

• \(|x - y| < \epsilon\)
• \(|y - z| < \epsilon\)

We need to check if this implies \(|x - z| < \epsilon\). Let's use the triangle inequality, which states that for any real numbers \(a\) and \(b\), \(|a + b| \le |a| + |b|\).

We can write \(x - z\) as \((x - y) + (y - z)\).

Using the triangle inequality:

\(|x - z| = |(x - y) + (y - z)| \le |x - y| + |y - z|\)

From our assumptions, we know \(|x - y| < \epsilon\) and \(|y - z| < \epsilon\).

So, \(|x - z| \le |x - y| + |y - z| < \epsilon + \epsilon = 2\epsilon\).

This inequality \(|x - z| < 2\epsilon\) does not guarantee that \(|x - z| < \epsilon\).

Let's consider a counterexample. Suppose \(\epsilon = 0.0005\). Let \(x = 0\), \(y = 0.0004\), and \(z = 0.0008\).

• Check \((x, y)\): \(|x - y| = |0 - 0.0004| = 0.0004\). Since \(0.0004 < 0.0005\), \((x, y) \in Re\).
• Check \((y, z)\): \(|y - z| = |0.0004 - 0.0008| = |-0.0004| = 0.0004\). Since \(0.0004 < 0.0005\), \((y, z) \in Re\).
• Check \((x, z)\): \(|x - z| = |0 - 0.0008| = |-0.0008| = 0.0008\). Since \(0.0008 \not< 0.0005\), \((x, z) \not\in Re\).

We found a case where \((x, y) \in Re\) and \((y, z) \in Re\), but \((x, z) \not\in Re\). Therefore, the relation \(Re\) is not transitive.

Summary of Properties

Conclusion based on Properties of Relation Re

Based on our analysis, the relation \(Re\) is Reflexive and Symmetric, but not Transitive.

Looking at the given options:

• (A) Reflexive: True
• (B) Symmetric: True
• (C) Transitive: False

We are looking for the option that correctly identifies which properties are true.

• Option 1: (A) and (B) only true - This matches our findings.
• Option 2: (B) and (C) only true - Incorrect, (C) is false.
• Option 3: (A) and (C) only true - Incorrect, (C) is false.
• Option 4: (A), (B) and (C) true - Incorrect, (C) is false.

Therefore, the correct option is the one stating that (A) and (B) only are true.

Revision Table: Properties of Relations

Additional Information: Equivalence Relations

A relation that is Reflexive, Symmetric, and Transitive is called an Equivalence Relation. Equivalence relations partition a set into disjoint subsets called equivalence classes.

The relation \(Re\) defined by \(|x - y| < \epsilon\) is a common example of a relation that is reflexive and symmetric but not transitive. Such relations are sometimes called tolerance relations or proximity relations. They capture the idea of elements being "close" or "approximately equal" within a certain tolerance \(\epsilon\).

The failure of transitivity in this specific case means that if \(x\) is close to \(y\) and \(y\) is close to \(z\), it doesn't necessarily mean \(x\) is close to \(z\) within the *same* small \(\epsilon\) threshold, as the accumulated difference can be larger than \(\epsilon\).