I have selfstudy on this subject and want to know if I am grasping the concept well. Here is the question:
Let $A=Z^+$, all integers that are positive; Let $R$= relation defined by $aRb$ iff there exists a $n$ in $Z^+$ such that $a=b^n$. Pick from the following what belongs to R.
i) $(1,9)$
ii) $(4,16)$
iii) $(5,5)$
iv) $(49,7)$
This is how I understand it:
(i) $a=1, b=9$ then $1=9^k$ hence $k=0$ but $0$ is not a member of $Z^+$ hence (i) does not belong to $R$.
(ii) $a=4, b=16$ then $4=16^k$ hence $k=1/2$ but k is not an element in $Z^+ ={1,2,3,4,...}$ hence (ii) does not belong to $R$. (4,16) cannot be.
(iii) $a=5,b=5$ then $5=5^k$ hence $k = 1$. 1 is member of set $Z^+$ hence (iii) belongs to $R$.
(iv) $a=49, b=7$ then $49=7^k$ hence $k = 2$ is a member of set $Z^+$. Therefore (iv) belongs to $R$.
Is this correct?