Question

Let $R$ be a relation from $N$ to $N$ defined by $R = \{(a, b) : a, b \in N$ and $a = b^2\}$. Which of the following is true?

• A. $(a, a) \in R$, for all $a \in N$
• B. $(d, b) \in R$, implies $(b, a) \in R$
• C. $(a, b) \in R$, $(b, c) \in R$ implies $(a, c) \in R$
• D. None of these

Answer 1

The Correct Option is D

We are given $R = \{(a, b): a, b \in N$ and $a = b^2\}$ $\{(b^2, b) : b \in N\}$ (a) False. True, only when $a = 1$, $(a, a) = (1,1) = (1^2, 1) \in R$. (b) False. If $\left(a,b\right) \in R$ $\Rightarrow a=b^{2} ? $\therefore \left(a, b\right) \in R$ $\Rightarrow \left(b, a\right) \notin R$. (c) False. If $\left(a, b\right) \in R$ $\Rightarrow a=b^{2}\quad\ldots\left(i\right)$ and $\left(b,c\right)\in R$ $\Rightarrow b=c^{2}\quad\ldots\left(ii\right)$ From $\left(i\right)$ and $\left(ii\right), a=\left(c^{2}\right)^{2}=c^{4} ? $\Rightarrow (a, b) \in R$ and $(b, c) \in R$ but $(a, c) \in R$.