Question

A relation \( R \) defined on a set of human beings as \( R = \{(x, y) : x \text{ is 5 cm shorter than } y\} \) is:

• A. Reflexive only
• B. Reflexive and transitive
• C. Symmetric and transitive
• D. Neither transitive, nor symmetric, nor reflexive

Hint:

Check the definitions of reflexive, symmetric, and transitive properties carefully for any given relation.

Answer 1

The Correct Option is D

Step 1: Check reflexivity. For reflexivity, \( (x, x) \) must belong to \( R \), but \( x \) cannot be 5 cm shorter than itself. Thus, \( R \) is not reflexive. 
Step 2: Check symmetry. For symmetry, if \( (x, y) \in R \), then \( (y, x) \in R \). Since \( x \) is 5 cm shorter than \( y \), the reverse is not true, so \( R \) is not symmetric. 
Step 3: Check transitivity. For transitivity, if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). However, \( x \) is 5 cm shorter than \( y \) and \( y \) is 5 cm shorter than \( z \), making \( x \) 10 cm shorter than \( z \). Thus, \( R \) is not transitive. 
Final Answer: \[ \boxed{\text{Neither transitive, nor symmetric, nor reflexive}} \]