# Determing If Relations Are Irreflexive

The question is, "Which relations in Exercise 5 are irreflexive?"

Exercise five being: Determine whether the relation R on the set of all Webpages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

a) everyone who has visited Web page a has also visited Webpage b.

b)there are no common links found on both Webpage a and Web page b.

c) there is at least one common link on Web page a and Webpage b.

d)there is a Web page that includes links to both Webpage a and Web page b.

I thought b,c, and d were irreflexive, but boy was I wrong. It turns out that none of them are irreflexive; I am having a difficult time seeing this, though. I could really use some help working through why they aren't irreflexive, please.

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A relation is reflexive if $(a,a) \in R$ for any $a$. If we just look for reflexivity, your examples become

• everyone who has visited webpage $a$ has also visited webpage $a$
• there are no common links found on both webpage $a$ and webpage $a$
• there is at least one common link on webpage $a$ and webpage $a$
• there is a webpage that includes links to both webpage $a$ and webpage $a$

The first, third, and fourth are reflexive (assuming the webpage contains links and is linked to in the first place). The second is not reflexive.

To incorporate Ross Millikan's comment, the problem is not completely specified.

Suppose the webpage $a$ contains no links. In particular, it will have no links in common with itself, and so $a$ will be related to itself in the second example, but not in the third. If webpage $a$ contains any links at all, however, then the situation is reversed.

If webpage $a$ is not linked to at all, then $a$ is not related to itself in the fourth example.

I feel like the intention of the problem was that all webpages contain links and all pages are linked to by some other webpage, but, strictly speaking, this need not be the case.

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For the third, suppose there is a page that has no links. For the fourth, it is conceivable there is a page that is not linked to anywhere. – Ross Millikan Nov 3 '12 at 1:04
Wow, you've answered my question, and were polite. This is so unlike the forms at MHF. I am glad I made the switch. Thank you for your help. – Mack Nov 3 '12 at 1:18
@EMACK Thank you for the kind words, and welcome to the community. – Austin Mohr Nov 3 '12 at 1:19