symmetric but not tranisitive nor reflexive function

Question

Let's say there is a function $g: B \rightarrow B$ and $B$ is some set.

A relation $Rx$ over set $B$ is when

$a Rx b$

if $g(a) = b$.

In this case, what kind of function $g: N \rightarrow N$ makes $Rx$ symmetric but not transitive nor reflexive?

I don't really understand this question. Does $g: N \rightarrow N$ mean the input is a natural number and output is also?

I know symmetry is when for elements of $B, b_1$ and$ b_2$, if $b_1$ is related to $b_2$ then $b_2$ is related to b1 but I was wondering how this would be applied here.

Thanks!

Answer 1

As far as I understand you are looking for the following kind of example.

let $f: N \rightarrow N $ be defined as follows $$f(x) = \text{smallest prime larger than }x -x$$ Then, if f(x)=y, then the relation $xRy$ is equivalently given as follows $$xRy \Leftrightarrow x+y = \text{prime number} $$ The above relation is symmetric but not reflexive or transitive.

Answer 2

Consider the following function (here I'm considering $0 \notin \mathbb{N}$, but it can be easily adapted to include $0$): $$f: \mathbb{N} \to \mathbb{N} $$ $$ f(n)= \begin{cases} n + 1&\text{if}\, n \text{ is odd}\\ n - 1&\text{if}\, n \text{ is even} \end{cases} $$

Then $f$, seen as a relation, is symmetric, but not reflexive or transitive.