Reflexive relations and the diagonal

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In a first-year FPM workshop last year, I mentioned the diagonal in the Cartesian square of a set, and the connection with reflexive relations on the set. Relations R on a set S correspond to subsets M of S×S in a standard way. And then the reflexive relations on S correspond to those subsets M of S×S such that the diagonal is a subset of M.

I had a question on Piazza asking me to explain a bit more about what the diagonal is. Here is my reply.

Hi,
For a set S you can form the Cartesian square S×S={(x,y):x,y∈S}. For example R×R={(x,y):x,y∈R}=R2. The diagonal (let's call it D) is a special subset of the Cartesian square S×S defined by D={(x,y)∈S×S:y=x}={(x,x):x∈S}. You can think of this as the graph of the identity map iS:S→S (where, for all x∈S, iS(x)=x).
If S=R, the diagonal is just the straight line through the origin in R2 given by the equation y=x.
If S is a finite set (for example, consider the special case S={1,5,9}), then S×S will also be a finite set (in our special case the Cartesian square has 32=9 elements) and the diagonal will be a finite set with the same number of elements as S (in our special case, the diagonal is just {(1,1),(5,5),(9,9)}).
Best wishes,
Dr Feinstein

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