Question

Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Answer 1

Reflexivity:

$$\begin{gathered} \mathrm{Let}\mathit{a}beanarbitraryelementof\mathrm{R.}Then, \\ a=a+1\mathrm{cannot}\mathrm{be}\mathrm{true}\mathrm{for}\mathrm{all}a\in A. \\ \Rightarrow \left(a,a\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{reflexive}\mathrm{on}A. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow b=a+1 \\ \Rightarrow -a=-b+1 \\ \Rightarrow a=b-1 \\ \mathrm{Thus},\left(b,a\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric}\mathrm{on}A. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(1,2\right)\mathrm{and}\left(2,3\right)\in R \\ \Rightarrow 2=1+1\mathrm{and}32+1\mathrm{is}\mathrm{true}. \\ \mathrm{But}3\ne 1+1 \\ \Rightarrow \left(1,3\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$