How can less-than-or-equal be "reflexive"?

Question

I just don't get how anything can be in relation to itself in any other way than being equivalent (being "the same"). How can some x be smaller/less than x? Intuitively, this makes no sense to me because if it were smaller it would not be in relation to itself, would it?

Answer 1

Let $E$ be a non-empty set. A relation on $E$ is simply a subset $\mathcal{R}$ of $E\times E$.

Let $x\in E$ and $y\in E.$ We want to relate $x$ and $y$ $\color{blue}[(x,y)\in \mathcal{R}$ in this order (hence the choice of $(\xi,\eta)$ and not $\{x,y\})\color{blue}]$ or we don't want to relate $x$ and $y$ $[(x,y)\notin \mathcal{R}]$.

For example, if $E=\{1,2,3,4,5\}$ :

• $\mathcal{R}_1 $ defined by :$(x,y)\in \mathcal{R}_1 \iff x+y=4$ is $$\mathcal{R}_1 =\{(1,3),(2,2),(3,1)\}$$
• $\mathcal{R}_2 $ defined by :$(x,y)\in \mathcal{R}_2 \iff x<y$ is $$\mathcal{R}_2 =\{(1,2),..., (4,5)\}$$
• $\mathcal{R}_3 $ defined by :$(x,y)\in \mathcal{R}_3 \iff x$ less-than-$\color{red}{or}$-equal $ y$ $$\mathcal{R}_2 =\{(1,1),..., (5,5)\}$$