Question

An integer m is said to be related to another integer n if m is a multiple of n.Check if the relation is symmetric, reflexive and transitive.

Answer 1

$$\begin{gathered} R=\left\{\left(m,n\right):m,n\in Z,m=kn,\mathrm{where}k\in N\right\} \\ \mathrm{Reflexivity}: \\ \mathrm{Let}m\mathrm{be}\mathrm{an}\mathrm{arbitrary}\mathrm{element}\mathrm{of}R.\mathrm{Then}, \\ m=km\mathrm{is}\mathrm{true}\mathrm{for}k=1 \\ \Rightarrow \left(m,m\right)\in R \\ \mathrm{Thus},R\mathrm{is}\mathrm{reflexive}. \\ \mathrm{Symmetry}:\mathrm{Let}\left(m,n\right)\in R \\ \Rightarrow m=kn\mathrm{for}\mathrm{some}k\in N \\ \to n=\frac{1}{k}m \\ \Rightarrow \left(n,m\right)\notin R \\ \mathrm{Thus},R\mathrm{is}\mathrm{not}\mathrm{symmetric}. \\ \mathrm{Transitivity}:\mathrm{Let}\left(m,n\right)\mathrm{and}\left(n,o\right)\in R \\ \Rightarrow m=kn\mathrm{and}n=lo\mathrm{for}\mathrm{some}k,l\in N \\ \Rightarrow m=\left(kl\right)o \\ \mathrm{Here},kl\in R \\ \Rightarrow \left(m,o\right)\in R \\ \mathrm{Thus},R\mathrm{is}\mathrm{transitive}. \end{gathered}$$