Question

The following relations are defined on the set of real numbers.
(i) aRb if a – b > 0
(ii) aRb if 1 + ab > 0
(iii) aRb if |a| ≤ b

Find whether these relations are reflexive, symmetric or transitive.

Answer 1

(i)
Reflexivity: Let a be an arbitrary element of R. Then,

$$\begin{gathered} a\in R \\ \mathrm{But}a-a=0\ngtr 0 \\ \mathrm{So},\mathrm{this}\mathrm{relation}\mathrm{is}\mathrm{not}\mathrm{reflexive}. \end{gathered}$$

Symmetry:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow a-b>0 \\ \Rightarrow -(b-a)>0 \\ \Rightarrow b-a<0 \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{relation}\mathrm{is}\mathrm{not}\mathrm{symmetric}. \end{gathered}$$

Transitivity:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R.\mathrm{Then}, \\ a-b>0\mathrm{and}b-c>0 \\ \mathrm{Adding}\mathrm{the}\mathrm{two},\mathrm{we}\mathrm{get} \\ a-b+b-c>0 \\ \Rightarrow a-c>0 \\ \Rightarrow \left(a,c\right)\in R. \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{relation}\mathrm{is}\mathrm{transitive}. \end{gathered}$$

(ii)
Reflexivity: Let a be an arbitrary element of R. Then,

$$\begin{gathered} a\in R \\ \Rightarrow 1+a\times a>0 \\ \mathrm{i}.\mathrm{e}.1+a^2>0\left[\mathrm{Since},\mathrm{square}\mathrm{of}\mathrm{any}\mathrm{number}\mathrm{is}\mathrm{positive}\right] \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{relation}\mathrm{is}\mathrm{reflexive}. \end{gathered}$$

Symmetry:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow 1+ab>0 \\ \Rightarrow 1+ba>0 \\ \Rightarrow \left(b,a\right)\in R \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{relation}\mathrm{is}\mathrm{symmetric}. \end{gathered}$$

Transitivity:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow 1+ab>0\mathrm{and}1+bc>0 \\ \mathrm{But}1+ac\ngtr 0 \\ \Rightarrow \left(a,c\right)\notin R \\ \mathrm{So},\mathrm{the}\mathrm{given}\mathrm{relation}\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$

(iii)
Reflexivity: Let a be an arbitrary element of R. Then,

$$\begin{gathered} a\in R \\ \Rightarrow \left|a\right|\nless a\left[\mathrm{Since},\left|a\right|=a\right] \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{reflexive}. \end{gathered}$$

Symmetry:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R \\ \Rightarrow \left|a\right|\le b \\ \Rightarrow \left|b\right|\nleqq a\mathrm{for}\mathrm{all}a,b\in R \\ \Rightarrow \left(b,a\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric}. \end{gathered}$$

Transitivity:

$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R\mathrm{and}\left(b,c\right)\in R \\ \Rightarrow \left|a\right|\le b\mathrm{and}\left|b\right|\le c \\ \mathrm{Multiplying}\mathrm{the}\mathrm{corresponding}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \left|a\right|\left|b\right|\le bc \\ \Rightarrow \left|a\right|\le c \\ \Rightarrow \left(a,c\right)\in R \\ \mathrm{Thus},R\mathrm{is}\mathrm{transitive}. \end{gathered}$$