Question

Test whether the following relations R₁, R₂, and R₃ are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R₁ on Q₀ defined by (a, b) ∈ R₁ ⇔ a = 1/b.
(ii) R₂ on Z defined by (a, b) ∈ R₂ ⇔ |a – b| ≤ 5
(iii) R₃ on R defined by (a, b) ∈ R₃ ⇔ a² – 4ab + 3b² = 0.

Answer 1

(i) Reflexivity:
Let a be an arbitrary element of R₁. Then,
$$\begin{gathered} a\in R_1 \\ \Rightarrow a\ne \frac{1}{a}\mathrm{for}\mathrm{all}a\in {\mathrm{Q}}_0 \\ \mathrm{So},R_1\mathrm{is}\mathrm{not}\mathrm{reflexive}. \end{gathered}$$

Symmetry:
Let (a, b) $\in R_1$. Then,

$$\begin{gathered} \left(a,b\right)\in R_1 \\ \Rightarrow a=\frac{1}{b} \\ \Rightarrow b=\frac{1}{a} \\ \Rightarrow \left(b,a\right)\in R_1 \\ \mathrm{So},R_1\mathrm{is}\mathrm{symmetric}. \end{gathered}$$

Transitivity:
Here,
$$\begin{gathered} \left(a,b\right)\in R_1\mathrm{and}\left(b,c\right)\in R_2 \\ \Rightarrow a=\frac{1}{b}\mathrm{and}b=\frac{1}{c} \\ \Rightarrow a=\frac{1}{{\displaystyle \frac{1}{c}}}=c \\ \Rightarrow a\ne \frac{1}{c} \\ \Rightarrow \left(a,c\right)\notin R_1 \\ \mathrm{So},R_1\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$

(ii)
Reflexivity:
Let a be an arbitrary element of R₂. Then,
$$\begin{gathered} a\in R_2 \\ \Rightarrow \left|a-a\right|=0\le 5 \\ \mathrm{So},R_1\mathrm{is}\mathrm{reflexive}. \end{gathered}$$

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R_2 \\ \Rightarrow \left|a-b\right|\le 5 \\ \Rightarrow \left|b-a\right|\le 5\left[\mathrm{Since},\left|a-b\right|=\left|b-a\right|\right] \\ \Rightarrow \left(b,a\right)\in R_2 \\ \mathrm{So},R_2\mathrm{is}\mathrm{symmetric}. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Let}\left(1,3\right)\in R_2\mathrm{and}\left(3,7\right)\in R_2 \\ \Rightarrow \left|1-3\right|\le 5\mathrm{and}\left|3-7\right|\le 5 \\ \mathrm{But}\left|1-7\right|\nleqq 5 \\ \Rightarrow \left(1,7\right)\notin R_2 \\ \mathrm{So},R_2\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$

(iii)
Reflexivity: Let a be an arbitrary element of R₃. Then,
_{$$\begin{gathered} a\in R_3 \\ \Rightarrow a^2-4a\times a+3a^2=0 \\ \mathrm{So},R_3\mathrm{is}\mathrm{reflexive}. \end{gathered}$$}

Symmetry:
$$\begin{gathered} \mathrm{Let}\left(a,b\right)\in R_3 \\ \Rightarrow a^2-4ab+3b^2=0 \\ \mathrm{But}{b}^2-4ba+3a^2\ne 0\mathrm{for}\mathrm{all}a,b\in R \\ \mathrm{So},R_3\mathrm{is}\mathrm{not}\mathrm{symmetric}. \end{gathered}$$

Transitivity:
$$\begin{gathered} \left(1,2\right)\in R_3\mathrm{and}\left(2,3\right)\in R_3 \\ \Rightarrow 1-8+6=0\mathrm{and}4-24+27=0 \\ \mathrm{But}1-12+9\ne 0 \\ \mathrm{So},R_3\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$