GATE CS 2009 Q4 — Sets, Relations and Functions

To determine the properties of the relation $R = \{(x,y), (x,z), (z,x), (z,y)\}$ on the set $\{x,y,z\}$, we test for symmetry and antisymmetry:

1. Symmetry: A relation $R$ is symmetric if for every $(a, b) \in R$, we also have $(b, a) \in R$.
   • Here, $(x, y) \in R$, but $(y, x) \notin R$.
   • Therefore, $R$ is not symmetric.
2. Antisymmetry: A relation $R$ is antisymmetric if for any $a \neq b$, if $(a, b) \in R$, then $(b, a) \notin R$.
   • Here, both $(x, z) \in R$ and $(z, x) \in R$, but $x \neq z$.
   • Therefore, $R$ is not antisymmetric.

Since $R$ is neither symmetric nor antisymmetric, the correct option is D.