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Relations, Set Theory, Reflexive, Symmetric, Transitive

  1. Aug 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Determine whether the relations on three sets are Reflexive, Irrelfexive, Symmetric,, Asymmetric, Antisymmetric, Transitive, and Intransitive.

    The relation [itex]\subseteq[/itex] on a set of sets.

    2. Relevant equations
    3. The attempt at a solution

    I am having trouble figuring out how this will work. I think for the set {{a},{b},{c}} that should mean

    <{a} , {{a},{b},{c}}>
    <{b} , {{a},{b},{c}}>
    <{{a},{b}} , {{a},{b},{c}}>
    <{{a},{b}} , {{a},{b},{c}}>
    <{{a},{b}} , {{a},{b},{c}}>
    <{{a},{b},{c}} , {{a},{b},{c}}>
    <∅ , {{a},{b},{c}}>

    But I do not know how to use this to answer the question.
     
  2. jcsd
  3. Aug 25, 2012 #2

    tiny-tim

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    Welcome to PF!

    Hi Cabinbreaker! Welcome to PF! :smile:

    (btw, you don't need those extra {} round most of your letters :wink:)
    eg reflexive means: if {a,b} then {b,a}

    transitive means: if {a,b} and {b,c} then {a,c} :smile:
     
  4. Aug 25, 2012 #3
    Then on the relation ⊆ on a set of sets (considering the set {{a},{b},{c}}), relations should be nonrelfexive, nonsymmetric, transitive, and Antisymetric.

    Relations are not irreflexive or intransitive.

    I think they are also not asymmetric because of <{{a},{b},{c}} , {{a},{b},{c}}> where xRy ⇒ ¬yRx ?
     
  5. Aug 25, 2012 #4

    vela

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    What you've written doesn't make much sense to me. Either I'm not understanding the question or you've misinterpreted it.

    Here's my take:

    You have S={{a}, {b}, {c}}, which is a set of sets, and let ##x, y \in S##. You say x is related to y if ##x\subset y##. For example, say x={a} and y={b}. It's not true that ##\{a\} \subseteq \{b\}##, so you would not say that {a} is related to {b}.

    In general, a relation R is a subset of S×S. In this case, S×S contains the nine ordered pairs

    {{a}, {a}}
    {{a}, {b}}
    {{a}, {c}}
    {{b}, {a}}
    {{b}, {b}}
    {{b}, {c}}
    {{c}, {a}}
    {{c}, {b}}
    {{c}, {c}}

    So those are the pairs you should be considering whereas you seem to be considering the power set of S. Can you state the problem exactly as given to you?
     
  6. Aug 25, 2012 #5
    The Problem says:

    My difficulty is I do not understand the question. I was able to complete the first two. I do not know what it means by "The relation ⊆ on a set of sets." I used the power set because I thought where A ⊆ B it was asking for the ordered pair {A,B}.
     
  7. Aug 25, 2012 #6

    vela

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    The set {{a}, {b}, {c}} is already a set of sets. That is, the elements of the set are sets themselves. You have to have a set of sets otherwise the relation ##\subseteq## doesn't make sense. For instance, if you have the set {a, b, c}, then you might ask if ##a \subseteq b##, which doesn't make sense because neither a nor b is a set. On the other hand, with the set {{a}, {b}, {c}}, one element is the set {a} and another is the set {b}, and it does make sense to ask if ##\{a\} \subseteq \{b\}##.

    I enumerated the nine ordered pairs in S×S. What you want to do first is determine if the first set in each pair is a subset of the second set in the pair. Then you want to determine if the relation is reflexive, etc. If the relation is reflexive, you should find that every pair of the form {{x}, {x}} satisfies the relation. And so on.
     
  8. Aug 25, 2012 #7

    HallsofIvy

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    Re: Welcome to PF!

    This is "symmetric", not "reflexive". A relation on set X is "reflexive" if an only if it contains {x,x} for every x in X.

     
  9. Aug 25, 2012 #8

    HallsofIvy

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    Then you do NOT want to look at a specific set as you do below.
    A relation on a given set is a collection of ordered pairs from that set. If we are given a set, X, A and B are two subsets the the pair (A, B) will be in the relation if and only if [itex]A\subseteq B[/itex].

    "Reflexive" here means that for any subset, A, of X, it is true that [itex]A\subseteq A[/itex] which is true because of the "=" part.
    "Irreflexive" simply means "not reflexive".

    "Transitive" here means "if [itex]A\subseteq B[/itex] and [itex]B\subseteq C[/itex] then [itex]A\subseteq C[/itex]".

    "Intransitive" simply means "not transitive".

    "Symmetric" here means "if [itex]A\subseteq B[/itex] then [itex]B\subseteq A[/itex]". Of course, that is not true.

    "Asymmetric" simply means "not symmetric".

    "Anti-symmetric" is a little more complicated. It means "if [itex]A\subseteq B[/itex] then it is NOT true that [itex]B\subseteq A[/itex]. Here that is not true because of the "=" part.

     
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