Question

Define a relation $R$ on $A =\\{1, 2, 3, 4\\}$ as $_xR_y$ if $x$ divides $y$. $R$ is

• A. reflexive and transitive
• B. reflexive and symmetric
• C. symmetric and transitive
• D. equivalence

Answer 1

Answer: (A)

reflexive and transitive

Answer 2

Given set $A=\\{1,2,3,4\\}$ and relation, $x R y$ if $x$ divides $y$. $\Rightarrow$ Relation $=\\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,3)(1,4),(2,4)\\}$ We have, $x R y \Leftrightarrow y / x$ for $x, y \in A$ For any $x \in A$, we have $x / x \Rightarrow x R x$ Thus, $x R x$ for all $x \in A .$ So, $R$ is reflexive on $A$. $R$ ia not symmetry because, if $y / x$, then $x$ may not divide $y .$ For example $4 / 2$ but $2 / 4$ Let $x, y, z \in A$, such that $x R y$ and $y R z .$ Then, $x R y$ and $y R z \Rightarrow \frac{y}{x}$ and $\frac{z}{y} \Rightarrow \frac{z}{x} .$ So, $R$ is a transitive relation on $A$.