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Clearly R is a symmetric relation.

Now for finding the set A, I don’t know the proper method to do it, but I managed to find $A=\{1,2,3,4\}$ (please let me know how to solve without trial and error)

Now, $(4,4)$ isn’t in the relation, so does that prohibit it from reflexive? I am just having a problem in the technicality of the question

Aditya
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  • Yes, you correctly found $A={1,2,3,4}$. Yes, since $4\in A$ but $(4,4)\notin R$ this implies that $R$ is not reflexive. Yes, this relation is symmetric. Now, all that commonly remains to be checked is if it is transitive. For that, I see $(1,2)$ is an element and $(2,3)$ is an element of $R$... – JMoravitz Mar 15 '21 at 12:56

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Note that the function $f(x) = 3^x - 4^{x-1}$ is eventually decreasing after some value of $x$. Since we are interested in only positive integer solutions and since $f(5) < 0$, your set $A$ is complete.

The relation $R$ is indeed symmetric but not reflexive and transitive. I am sure you can find out why.

VIVID
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  • Actually R not being reflexive is the second part of my question – Aditya Mar 15 '21 at 13:13
  • @Aditya $R$ is not reflexive, since $\forall x \in A \not\Rightarrow (x,x) \in R$. Consider $4 \in A$ but $(4,4) \notin R$. – VIVID Mar 15 '21 at 13:18