Let $R$ be a relation from $N$ to $N$ defined by $R = \{(a, b) : a, b \in N \text{ and } a = b^2\}$. Is the following statement true?
$(a, b) \in R$ implies $(b, a) \in R$
$(a, b) \in R$ implies $(b, a) \in R$
(B) Given $R = \{(a, b) : a, b \in N \text{ and } a = b^2\}$.
To check if $(a, b) \in R \implies (b, a) \in R$,we test a counterexample.
Consider the pair $(9, 3)$. Since $9 = 3^2$ and $9, 3 \in N$,we have $(9, 3) \in R$.
Now,check if $(3, 9) \in R$. For $(3, 9)$ to be in $R$,it must satisfy $a = b^2$,which means $3 = 9^2 = 81$. Since $3 \neq 81$,$(3, 9) \notin R$.
Therefore,the statement $(a, b) \in R \implies (b, a) \in R$ is false.
To check if $(a, b) \in R \implies (b, a) \in R$,we test a counterexample.
Consider the pair $(9, 3)$. Since $9 = 3^2$ and $9, 3 \in N$,we have $(9, 3) \in R$.
Now,check if $(3, 9) \in R$. For $(3, 9)$ to be in $R$,it must satisfy $a = b^2$,which means $3 = 9^2 = 81$. Since $3 \neq 81$,$(3, 9) \notin R$.
Therefore,the statement $(a, b) \in R \implies (b, a) \in R$ is false.
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