[new thread since “irreflexive relation” was not found among the LatestChanges threads]

A few day ago I added a standout box to irreflexive relation suggesting clarifying a notation.

I did this since there seems to be something to be clarified, but there is, as far as I can tell, nothing more to do than replace $x\nsim x$ with $\neg\, (x\sim x)$, and

- I thought the standout-box-route to be the most efficient and silentest.

Since this appears not to have worked out, this message.

Again, it seems that $\nsim$ is nothing else than an abbreviation, definitionally-equal to $\neg\, (x\sim x)$. It seems to be that this should be spelled out, the $\nsim$ not being defined anywhere (definitely not on the page itself, and I looked around a bit), and it is at this point most probably *not* meant to denote an apartness-relation, distinct from the relation $\sim$, i.e., $\nsim$ is not a relation symbol, in other words, *not part of the syntax*, rather part *of a meta-syntax*.

Consider sets and functions. A relation between two sets can then be expressed as a subset of a Cartesian product, in other words, we can define it and describe it using functions.

Viceversa, can we describe functions using relations as the “fundamental” arrows? That is, can we define and describe functions without using other functions, only relations?

]]>