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just want to verify that my understanding of relations is correct, grammar and correct logical form. Thanks!

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where $(a, b) \in R$ if and only if every one who has visited Web page a has also visited Web page b.

Reflexive: One who has visited Web page a has also visited Web page a. Thus this is reflexive.

Symmetric: One who visits Web page a has also visited web page b, however it is not necessarily the case that one who visits Web page b has visited Web page a, thus this is not symmetric.

Transitive: If one has visited Web page a, they visited Web page b, and having visited Web page b they also visited Web page c. Thus, one who has visited Web page a has visited Web page c. Thus this relation is transitive.

Antisymmetric We know one who visits Web page a, also visits Web page b, from this if we know that one who visits Web page b also visits Web page a, then Web page a is Web page b. Thus, this relationship is antisymmetric.

javaderek
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  • You're right about everything except antisymmetry. Just because two sites have exactly the same visitors doesn't mean they're the same site. – BrianO Mar 12 '16 at 22:31

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The first three seem correct to me, but the last one does not: there may be two different websites that happen to have been visited by precisely the same users. So $(a, b)\in R$ and $(b, a)\in R$ does not imply $a=b$ in general, in which case $R$ is not antisymmetric.

Damian Reding
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  • I think your right, I didn't apply the definition of antisymmetric properly, like you said it would have to be the case: "all users who visits web page a visit web page b, and all users who visit web page b visit web page a", does not mean that they are the same web page. Thank you! – javaderek Mar 12 '16 at 21:38
  • You're welcome! – Damian Reding Mar 12 '16 at 21:41
  • I have only one more question, regarding the question write up, it states "if and only if", have I responded correctly with my solutions? Would I need to now assume a position of reflexive, symmetric, antisymmetric and transitive? Or is the question incomplete? – javaderek Mar 12 '16 at 22:07
  • Well, for my taste your write-up is fine, I don't know about your teacher.. – Damian Reding Mar 12 '16 at 22:15