This is a question from "Discrete Mathematics and Its Applications":
Determine whether the relation R on the set of all Web
pages is reflexive, symmetric, antisymmetric, and/or tran-sitive,
where(a, b) ∈ R if and only if
a) everyone who has visited Web page a has also visited Web page b .
The answer is:
a) Reflexive, transitive
I don't understand well why it's "Reflexive". If there is a web page which has never been visited by anybody, (a,a) won't be belonged to R, that R should not be reflexive, right?
Do I misunderstand something wrong?
Thanks for @Brian M. Scott's answer, I understand it now. Let me add something important here:
The description of $R$ is:
$R$={$(x,y) | \forall p (F(p,x) \to F(p,y))$}, where $F(p,x)$ means "p has visited x", and $ p \in P $ where P is "all people".
When page $x$ has never been visited, that $F(p,x), p \in P$ is always false, that $\forall p (F(p,x) \to F(p,y))$ is always true, that F(x, any y) is always true.
That's why the relation is "Reflexive".
If $a$ is a web page, let $V(a)$ be the set of people who have visited $a$. Then $\langle a,b\rangle\in R$ if and only if $V(a)\subseteq V(b)$. Now it’s always true that $V(a)\subseteq V(a)$, even if $V(a)=\varnothing$, so it’s always true that $\langle a,a\rangle\in R$. Thus, $R$ really is reflexive.
Here’s a slightly different way to look at it. If $a$ and $b$ are particular web pages, how could you prove that $\langle a,b\rangle\notin R$? You would have to find someone who has visited $a$ but has not visited $b$. If no one has visited $a$, you can’t do this: you can’t find anyone who has visited $a$, let alone someone who has visited $a$ but not $b$! This shows that if $a$ has never been visited, then $\langle a,b\rangle\in R$ for all $b$ (and hence certainly for the particular case $b=a$).