I've found the two textbooks I'm using to to be particularly unhelpful in explaining these concepts, especially as they relate to English examples (non-existent).
The first few following questions have answers provided, but I would like to understand why they are correct.
Determine whether the relation R on the set of all Web pages 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 Web page b.
correct answer: reflexive, transitive
so R: {(a,b) | everyone who has visited a has also visited b}
I thought this was anti-symmetrical because it's not necessarily the case that everyone who has visited b has also visited a. That is, we have (a,b) but not (b,a). But I see now maybe it is impossible to actually say anything about (b,a) given this piece of information?
I then consider it may be reflexive because everyone who has visited a has also visited a (ie., a = a, b = b, and the same would hold for all web pages c, d, and so on). Am I understanding this properly? If this is true, does transitivity stem form the fact that we have (a,a) and (a,b)?
The next three:
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 Web page b.
d) there is a Web page that includes links to both Web page a and Web page b.
correct answer: symmetrical for each
I have interpreted these to be symmetrical because every given relation R here shows (a,b) = (b,a), where the relation is no common links, at least one common link, or at least one webpage linking them both. Because the relations say nothing about pages linking to themselves (reflexivity), page a linking to b but not b linking a (anti-symmetry), or lack of any given instance like (a,b) and (a,a) or (a,b) (b,c) and (a,c) (transitivity).
These questions do not have answers, but I would like to know if my reasoning here is sound, to make sure I'm understanding what is going on. The questions are similar to above, given a relation on the set of all people.
a) a is taller than b.
I believe this is anti-symmetrical because a is taller than b, but it's not the case (and would be impossible) that b is taller than a.
b) a and b were born on the same day.
I believe this is symmetrical because (a,b) = (b,a), where the relation is their birthdays, which are the same.
c) a has the same first name as b.
Symmetrical for similar reasons above (the same first name).
d) a and b have a common grandparent.
Again, symmetrical due to their common grandparent.
Thank you for your help!
Best Answer
a) There is a difference between antisymmetrical and asymmetrical. If '$a$ taller than $b$' implies here that $a$ and $b$ have not the same length then the relation is not antisymmetrical but asymmetrical. In fact if: $$\left(a,b\right)\in R\wedge\left(b,a\right)\in R\Rightarrow a=b$$ then $R$ is antisymmetrical. If:
$$\left(a,b\right)\in R\Rightarrow\left(b,a\right)\notin R$$ then $R$ is asymmetrical
b,c,d) Here $a$ and $b$ can be switched in the sentence without touching its meaning. That tells us that the relation is symmetrical.