The question is, "Show that the relation R = ∅ on the empty set S = ∅ is reflexive, symmetric, and transitive."
I was told by my teacher that you could simply say it can't be shown that each property isn't true; and that would show that the relation had those three properties. To me, this answer isn't very satisfying. Could someone, perhaps, elaborate on this idea more?
Thank you!
Best Answer
To show reflexivity, note that for every $x\in\varnothing$, we have $xRx$.
To show symmetry, note that for every $x,y\in\varnothing$, we have $xRy$ implies $yRx$.
To show transitivity, note that for every $x,y,z\in\varnothing$, we have $xRy$ and $yRz$ implies $xRz$.
These are vacuously true because the empty set contains no elements.