9. A binary relation $R$ defined on a set $S$ is said to be antisymmetric if for $x , y \in S , x R y$ and $y R x \Longrightarrow x = y$. Let $R _ { 1 } , R _ { 2 }$ be two binary relations defined on a set $S$. The union of $R _ { 1 } , R _ { 2 }$ is the binary relation $U$ defined on $S$ as: For $x , y \in S , x U y \Longleftrightarrow x R _ { 1 } y$ or $x R _ { 2 } y$. The intersection of $R _ { 1 } , R _ { 2 }$ is the binary relation $I$ defined on $S$ as: For $x , y \in S , x I y \Longleftrightarrow x R _ { 1 } y$ and $x R _ { 2 } y$. Which of the following statements is/are true? (a) A binary relation cannot be both symmetric and antisymmetric. (b) A binary relation can be both transitive and antisymmetric. (c) The union of two equivalence relations is always an equivalence relation. (d) The intersection of two equivalence relations is always an equivalence relation.
9. A binary relation $R$ defined on a set $S$ is said to be antisymmetric if for $x , y \in S , x R y$ and $y R x \Longrightarrow x = y$. Let $R _ { 1 } , R _ { 2 }$ be two binary relations defined on a set $S$. The union of $R _ { 1 } , R _ { 2 }$ is the binary relation $U$ defined on $S$ as: For $x , y \in S , x U y \Longleftrightarrow x R _ { 1 } y$ or $x R _ { 2 } y$. The intersection of $R _ { 1 } , R _ { 2 }$ is the binary relation $I$ defined on $S$ as: For $x , y \in S , x I y \Longleftrightarrow x R _ { 1 } y$ and $x R _ { 2 } y$.\\
Which of the following statements is/are true?\\
(a) A binary relation cannot be both symmetric and antisymmetric.\\
(b) A binary relation can be both transitive and antisymmetric.\\
(c) The union of two equivalence relations is always an equivalence relation.\\
(d) The intersection of two equivalence relations is always an equivalence relation.\\