Reflexivity of Relations

Question

Explain why the relation $$T=\{(a,b)\in\mathbb{R}\times\mathbb{R}:a≤b\}$$ is reflexive, but the relation $$ Q=\{(a,b)\in \mathbb{R}\times\mathbb{R}:a<b\}$$ is not reflexive.

Is $T$ reflexive because $a$ can equal $b$? While in $Q$, $a$ can only be smaller than $b$ and not equal to be reflexive as in $T$.

Answer 1

You've got it precisely right. We know that an equivalence relation $\sim$ on a set $X$ is reflexive if $x \in X$ satisfies $x\sim x$. Is it possible for a real number $x$ to satisfy $x<x?$ No. However, any real number can (and does) satisfy $x\le x$. So, $"\le"$ is a reflexive relation on $\mathbb{R}$, while $"<"$ is not.

Answer 2

If every element in the set is related to itself, then the relation is Reflexive.

every $a \in \mathbb{R}$ satisfies $a \leq a$. So $T=\{(a,b)\in\mathbb{R}\times\mathbb{R}:a≤b\}$ is reflexive.

No real number satisfies $a < a$. So $Q=\{(a,b)\in \mathbb{R}\times\mathbb{R}:a<b\}$ is not reflexive.