Practicing Success
Let S be the set of all real numbers. Then, the relation $R= \{(a, b): 1+ ab > 0\}$ on S is |
Reflexive and symmetric but not transitive Reflexive and transitive but not symmetric Symmetric and transitive but not reflexive None of the above is true |
Reflexive and symmetric but not transitive |
The correct answer is Option (1) → Reflexive and symmetric but not transitive We observe the following properties: Reflexivity: Let a be an arbitrary element of R. Then, $a ∈ R$ $⇒ 1+ a. a=1+ a^2 > 0$ [$∵ a^2>0$ for all $a ∈ R$] $⇒(a, a) ∈ R_1$ [By def. of $R_1$] Thus, $(a, a) ∈ R_1$ for all $a ∈ R$. So, $R_1$ is reflexive on R. Symmetry: Let $(a, b) ∈ R$. Then, $(a, b) ∈ R_1$ $⇒ 1+ ab > 0$ $⇒ 1 + ba > 0$ [$∵ ab = ba$ for all $a, b ∈ R$] $⇒ (b, a) ∈ R_1$ [By def. of $R_1$] Thus, $(a, b) ∈ R_1⇒ (b, a) ∈ R_1$ for all $a, b ∈ R$. So, $R_1$ is symmetric on R. Transitivity: We observe that $(1, 1/2) ∈ R_1$ and $(1/2, -1) ∈ R_1$ but $(1, -1)∉ R_1$ because $1 + 1 × (−1) = 0≯0$. So, $R_1$ is not transitive on R. |