A relation \( R \) defined on a set of human beings as \( R = \{(x, y) : x \text{ is 5 cm shorter than } y\} \) is:
Show Hint
- Reflexive only
- Reflexive and transitive
- Symmetric and transitive
- Neither transitive, nor symmetric, nor reflexive
The Correct Option is D
Solution and Explanation
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: \[ \boxed{\text{Neither transitive, nor symmetric, nor reflexive}} \]
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