How to write connected relation $\mathcal{R}$ in terms of set notation only.

Question

Definition: A relation $\mathcal{R}$ on set $X$ is called connected if for all $x,y\in X,\;$$x\mathcal{R} y$ or $y\mathcal{R} x$ or $x=y$

Let $\mathcal{R}$ be a connected relation on set $X$ and set $Y$, and write $\mathcal{R}$ only in terms of set notation.

I know that relation $\mathcal{R}$ is a subset of the cartesian product of $X\times Y$, but I have no idea how to precisely capture the $\color{red}{\text{connected}\;\mathcal{R} }$

The general set notation of relation I familiar with is $\mathcal{R}\subseteq X\times Y=\{(x,y): x\in X, y\in Y\}$

Please help

Answer 1

Let me preface my answer by correcting what might be a misconception of the OP, or at the very least a "mis-expression". The definition of a connected relation on $X$ requires a relation on $X$. So before one can even speak sensibly about connectivity of a relation between two sets $X$ and $Y$, one requires that $X=Y$. In fact the concept of a connected relation between two unequal sets is not defined.

With that in mind, you can easily do this if you make use of some standard notation:

• Cartesian product notation $X \times X$
• The diagonal subset $\Delta = \{(x,x) \mid x \in X\}$
• The inversion operator: $\mathcal R^{-1} = \{(x,y) \in X \times X \mid (y,x) \in \mathcal R \}$

It follows that $\mathcal R$ is connected if and only if $$\mathcal R \cup \mathcal R^{-1} \cup \Delta = X \times X $$