Question

If \[ R = \{ (x, y) : x \text{ is exactly } 7\text{cm taller than } y \} \] then \( R \) is:

• A. Not symmetric
• B. Reflexive
• C. Symmetric but not transitive
• D. An equivalence relation

Hint:

A relation is symmetric if \( (x, y) \Rightarrow (y, x) \), transitive if \( (x, y) \) and \( (y, z) \Rightarrow (x, z) \), and reflexive if \( (x, x) \) always holds.

Answer 1

The Correct Option is A

Step 1: Checking reflexivity. 
A relation is reflexive if \( (x, x) \) is in \( R \) for all \( x \). Since no one can be 7 cm taller than themselves, \( R \) is not reflexive. 
Step 2: Checking symmetry. 
A relation is symmetric if \( (x, y) \in R \) implies \( (y, x) \in R \). Since \( x \) is 7 cm taller than \( y \), but \( y \) is not 7 cm taller than \( x \), the relation is not symmetric. 
Step 3: Checking transitivity. 
A relation is transitive if \( (x, y) \in R \) and \( (y, z) \in R \) imply \( (x, z) \in R \). If \( x \) is 7 cm taller than \( y \) and \( y \) is 7 cm taller than \( z \), then \( x \) is 14 cm taller than \( z \), so \( R \) is not transitive. Thus, the correct answer is (A).