### Example: Proving a New Metric**Problem Statement:**Suppose \( d \) is a metric on a set \( X \). Show that \( \rho: X^2 \rightarrow [0, \infty) \) defined by\[ \rho(x, y) = \frac{3d(x, y)}{2 + 3d(x, y)} \]is also a metric on \( X \).**Explanation and Proof:**To show that \( \rho(x, y) \) is a metric on \( X \), we must verify that it satisfies the following four properties of a metric:1. **Non-negativity:** \( \rho(x, y) \geq 0 \).2. **Identity of indiscernibles:** \( \rho(x, y) = 0 \iff x = y \).3. **Symmetry:** \( \rho(x, y) = \rho(y, x) \).4. **Triangle inequality:** \( \rho(x, z) \leq \rho(x, y) + \rho(y, z) \).Let’s proceed with these in order:1. **Non-negativity:** - Since \( d(x, y) \) is a metric, it is always non-negative (\( d(x, y) \geq 0 \)). - The fraction \( \frac{3d(x, y)}{2 + 3d(x, y)} \) is composed of non-negative terms. Thus, \( \rho(x, y) \geq 0 \).2. **Identity of indiscernibles:** - When \( x = y \), \( d(x, y) = 0 \). - Therefore, \( \rho(x, y) = \frac{3 \cdot 0}{2 + 3 \cdot 0} = 0 \). - Conversely, if \( \rho(x, y) = 0 \), then \(\frac{3d(x, y)}{2 + 3d(x, y)} = 0 \). - This implies \(3d(x, y) = 0 \implies d(x, y) = 0 \). - Since \( d \) is a metric, \( d(x, y) = 0 \iff x = y \). Therefore, \( \

Name: ### Example: Proving a New Metric**Problem Statement:**Suppose \( d \
Uploaded: 2024-03-05T11:52:21.000Z
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Description: ### Example: Proving a New Metric**Problem Statement:**Suppose \( d \) is a metric on a set \( X \). Show that \( \rho: X^2 \rightarrow [0, \infty) \) defined by\[ \rho(x, y) = \frac{3d(x, y)}{2 + 3d(x, y)} \]is also a metric on \( X \).**Explanatio