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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
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.
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
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.
GATECSE