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Return to GATE2009 q4.

Consider the binary relation $R = \{(x,y), (x,z), (z,x), (z,y)\}$ on the set $\{x,y,z\}$. Which one of the following is TRUE?

(A) $R$ is symmetric but NOT antisymmetric

(B) $R$ is NOT symmetric but antisymmetric

(C) $R$ is both symmetric and antisymmetric

(D) $R$ is neither symmetric nor antisymmetric

Solution by Happy Mittal[edit]

A relation is symmetric if, for each $(x,y)$ in $R$, $(y,x)$ is also in $R$. This relation $R$ in question is not symmetric, because for $(x,y)$, $R$ doesn't have $(y,x)$.
A relation is antisymmetric if, for each pair of $(x,y)$ and $(y,x)$ in $R$, $x=y$. Here $R$ in question is not antisymmetric because for pairs $(x,z)$ and $(z,x)$, $x ≠ z$.
So, option (D) is correct.




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