Verify that any two elements of $X$ are connected by the relation $\mathcal{R}\subset X^{2}$ iff $\mathcal{R}\cup\mathcal{R}' = X^{2}$.

Question

The relation $\mathcal{R}'\subset Y\times X$ is called the transpose of the relation $\mathcal{R}\subset X\times Y$ if $(y\mathcal{R}'x)\Leftrightarrow (x\mathcal{R}y)$. Verify that any two elements of $X$ are connected (in some order) by the relation $\mathcal{R}\subset X^{2}$ if and only if $\mathcal{R}\cup\mathcal{R}' = X^{2}$.

MY ATTEMPT

I am not sure about how to interpret the expression "in some order", which disables me to solve the problem. Can someone help me out? Thanks in advance!

Answer 1

$x$ and $y$ are connected in some order if and only if $xRy$ or $yRx$ holds.