Question

Consider the following statements: P: \( d_1(x,y) = \left| \log \left( \frac{x}{y} \right) \right| \) is a metric on \( (0, 1) \). 
Q: \( d_2(x, y) = \begin{cases} |x| + |y|, & \text{if } x \neq y \\ 0, & \text{if } x = y \end{cases} \) is a metric on \( (0, 1) \). Then:

• A. both P and Q are TRUE
• B. P is TRUE and Q is FALSE
• C. P is FALSE and Q is TRUE
• D. both P and Q are FALSE

Hint:

To verify if a function is a metric, check the four properties: non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Answer 1

The Correct Option is A

Step 1: Analyze statement P.
For \( d_1(x,y) = \left| \log \left( \frac{x}{y} \right) \right| \) to be a metric on \( (0, 1) \), it must satisfy the four properties of a metric: 1. Non-negativity: \( d_1(x,y) \geq 0 \). 2. Identity of indiscernibles: \( d_1(x,y) = 0 \) if and only if \( x = y \). 3. Symmetry: \( d_1(x,y) = d_1(y,x) \). 4. Triangle inequality: \( d_1(x,z) \leq d_1(x,y) + d_1(y,z) \). For the function \( d_1(x,y) = \left| \log \left( \frac{x}{y} \right) \right| \), all these conditions are satisfied, so statement P is TRUE.

Step 2: Analyze statement Q.
For \( d_2(x, y) = \begin{cases} |x| + |y|, & \text{if } x \neq y
0, & \text{if } x = y \end{cases} \), we check the four metric properties. This function satisfies all conditions of a metric, including non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Thus, statement Q is also TRUE.

Final Answer: (A) both P and Q are TRUE