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Math Help - Determining Properties Of A Relation

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    Determining Properties Of A Relation

    The problem is: "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.

    I can see how it is transitive--it is always true that someone who visited webpage a also visited webpage a, or in ordered pair notation, (a, a).

    What I am having difficulty seeing is how it is not symmetric. Wouldn't it be true that, if you visited webpage a, then you visited webpage b, could be stated as, if you visited webpage b, then you visited webpage a? Meaning that (a, b) and (b, a) are elements of R?
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    Re: Determining Properties Of A Relation

    Quote Originally Posted by Bashyboy View Post
    The problem is: "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.
    I can see how it is transitive--it is always true that someone who visited webpage a also visited webpage a, or in ordered pair notation, (a, a).

    What I am having difficulty seeing is how it is not symmetric.
    The difficulty is the everyone.
    Is it necessarily true that everyone who visits B also visits A?
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    Re: Determining Properties Of A Relation

    Presumably, all Internet users visited Google. So all people who visited my home page also visited Google. But what about the converse?
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    Re: Determining Properties Of A Relation

    everyone who has visited this page (Determining Properties Of A Relation) henceforth know as "a", has presumably also visited the parent page: (Discrete Math) henceforth know as "b" (technically, it's possible to navigate directly to this page, but let's not quibble. you can conceive of a website where you HAVE to go to the main site (the "index page"), to get to a particular page on that site (one that might be password-protected, for example, so you have to go through the "log-in page" to get to any other page).

    therefore we have aRb.

    however, someone may have visited the parent forum of Discrete Math, without ever having looked at this particular topic, so we do not have bRa.

    in other words, sure, we know that the people who looked at a, also looked at b. but surely there can exist people who looked at just b, but never at a. and of course, these people count as part of "everyone".

    in fact, any kind of "gated access" leads to this sort of thing: if you have to go through checkpoint A to get to checkpoint B, then anyone who got as far as B made it through A, but we cannot say anyone who made it through A, will also make it past B. any kind of travel (including web-page clicking) might be "one-way" and is not, therefore, symmetric (which implies a kind of "two-way-ness").

    the grand-daddy of all non-symmetric (in fact ANTI-symmetric) relations is "≤", which arises in hierarchical structures of any kind (like sets, or web-pages, or organizations such as the military). there is a hidden appeal to such a kind of structure any time one uses something like "A implies B" (saying John is a man is pretty much saying the set of all men includes John). it is rarely the case that we can do the REVERSE thing, and conclude from the fact that someone is a man, that it must be John.

    in older language, if someone is John, he is necessarily a man (let's ignore all the weird exceptions of parents who chose to name their daughters "John", ok?), but it is not sufficient to say that if someone is a man, he is then John (that is identifying someone as "a man" is not sufficient information to deduce the man is "John").

    with equivalences, we are essentially saying: it makes no difference (for the purpose of "whatever") if we use A or B. we're relaxing the "strictness" of equality, but keeping its "rules" (equivalence extends equality (this is what reflexive means), equivalence is bi-directional (this is what symmetry means), and equivalent to equivalent is equivalent (this is what transitive means)).
    Last edited by Deveno; November 2nd 2012 at 10:21 AM.
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