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 \(S\times S\) in a standard way. And then the reflexive relations on \(S\) correspond to those subsets \(M\) of \(S\times S\) 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 \[S \times S=\{(x,y):x,y \in S\,\}\,.\] For example \[\mathbb{R}\times \mathbb{R}= \{(x,y):x,y \in \mathbb{R}\}=\mathbb{R}^2\,.\] The diagonal (let's call it \(D\)) is a special subset of the Cartesian square \(S \times S\) defined by \[D=\{(x,y)\in S \times S:y=x\,\} = \{(x,x): x\in S\}\,.\] You can think of this as the graph of the identity map \(i_S:S\to S\) (where, for all \(x \in S\), \(i_S(x)=x\)).
If \(S=\mathbb{R}\), the diagonal is just the straight line through the origin in \(\mathbb{R}^2\) given by the equation \(y=x\).
If \(S\) is a finite set (for example, consider the special case \(S=\{1,5,9\}\)), then \(S \times S\) will also be a finite set (in our special case the Cartesian square has \(3^2=9\) 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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