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Relations Again

  1. Feb 11, 2006 #1
    Relations Again :(

    K, so im studying for the upcoming midterm... and their is this question in the book...

    Let A = {a} and B = {1,2,3}. List all the possible relations between A and B.

    So Ordered pairs, Cartesian Products and Relations are all together in the chapter, and im really confused between them.

    "List all possible relations between A and B"

    Would that be something like A~B ?

    would the answer be something like {(a,1) (a,2) (a,3)} ?

    Please help someone!

    Thanks
     
  2. jcsd
  3. Feb 12, 2006 #2

    matt grime

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    Since you have the book around why don't you start with the thing that states

    Definition: a relation on two sets A and B is a.....

    Try writing it out here and starting to work out all possible cases and see where you get. People can then point out what you've done right and what you've missed out.
     
  4. Feb 12, 2006 #3
    Let A = {a} and B = {1,2,3}. List all the possible relations between A and B.

    Definition of a relation: Let A and B be sets. A relation between A and B is any subset R of AxB. We say that (a is in A) and (b is in B) are related by R if ((a,b) is in R), and we often denote this by writing "aRb."

    Given this defintion...
    A = {a}
    B = {1,2,3}

    AxB = {(a,1) (a,2) (a,3)}

    and so, all the possible relations are {(a,1) (a,2) (a,3)}
     
  5. Feb 12, 2006 #4

    matt grime

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    You think that AxB has exactly one subset? Does that seem at all reasonable? It doesn't to me. Only the empty set has exactly one subset in my experience. Try writing out another subset of AxB.
     
  6. Feb 12, 2006 #5
    hmmm...

    Definition: AxB= {(a,b): a in A and b in B}

    A= {a}
    B = {1,2,3}

    Therefore, the possiblities for AxB with the definition are:

    AxB = {(a,1)}
    AxB = {(a,2)}
    AxB = {(a,3)}

    Am i still missing something or doing something wrong?
     
  7. Feb 12, 2006 #6

    Galileo

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    AxB= {(a,b): a in A and b in B} is the correct definition.
    It means: "The set of all ordered pairs with the first component in A and the second in B."
    It doesn't mean a set consisting of one ordered pair with the first component in A and the second in B.

    So from this definition, write down what AxB is.
    Then look again at the definition of a relation and write down all possible relations between A and B.
     
  8. Feb 12, 2006 #7

    Hurkyl

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    Matt's suggestion is recursive. Do it again to the word subset.
     
  9. Feb 12, 2006 #8
    Lets look at another example for a moment.

    Let A = {1,2}. list all the ordered pairs (x,y) such that (x in A) and (y in A)

    Answer: (1,1) (1,2) (2,1) (2,2), I see 4 Ordered Pairs here.

    Let A = {1,2}. list all the ordered pairs (x,y) such that (x in A) and (y in B)

    Answer: (1,2) (2,1), I see 2 Ordered Pairs here.

    Back to the quesiton: Let A = {a} and B = {1,2,3}. List all the possible relations between A and B.

    Definition: A relation between A and B is any subset R of AxB
    Definition: AxB= {(a,b): a in A and b in B}

    AxB = {(a,1) (a,2) (a,3)} I see three Ordered Pairs in this set. I have no idea what im doing wrong here? i mean, (a in A) and ((1,2,3) in B) give these ordered pairs (a,1), (a,2), (a,3)

    Or perhaps its {(1,a) (2,a) (3,a} but that makes no sense to me nor to the definition.
     
  10. Feb 12, 2006 #9

    HallsofIvy

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    Where did B come from? Did you mean "(y in A)"?

    [quoteBack to the quesiton: Let A = {a} and B = {1,2,3}. List all the possible relations between A and B.

    Definition: A relation between A and B is any subset R of AxB
    Definition: AxB= {(a,b): a in A and b in B}

    AxB = {(a,1) (a,2) (a,3)} I see three Ordered Pairs in this set. I have no idea what im doing wrong here? i mean, (a in A) and ((1,2,3) in B) give these ordered pairs (a,1), (a,2), (a,3)

    Or perhaps its {(1,a) (2,a) (3,a} but that makes no sense to me nor to the definition.[/QUOTE]

    Yes, you are correct. AxB= {(a,1), (a, 2), (a, 3)} ({(1,a),(2,a),(3,a)}=
    BxA which is different) has three members. Now how many subsets does it have?
     
  11. Feb 12, 2006 #10
    Where did B come from? Did you mean "(y in A)"?

    Ooops... i made a mistake with the "(y in A)" ... forget about that!

    How many subsets does AxB have? Ill list all the possible subsets of AxB:

    C = {(a,1)}
    D = {(a,2)}
    E = {(a,3)}
    F = {(a,1),(a,2)}
    G = {(a,2),(a,3)}
    H = {(a,1),(a,3)}
    I = {(a,1), (a, 2), (a, 3)}

    So their are 7 possible subsets of AxB. So i have been confused all this time because matt grimm said in post #4 "You think that AxB has exactly one subset"
     
  12. Feb 12, 2006 #11

    Hurkyl

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    You missed a subset.


    He said that because when you wrote down what you claim was all relations, you wrote down one thing. And since a relation is a subset of AxB, it's as if you are asserting that AxB only has one subset.
     
  13. Feb 12, 2006 #12
    I suppose I missed the empty set

    J = {empty set}

    Question: Let A = {a} and B = {1,2,3}. List all the possible relations between A and B.

    A relation between A and B is any subset R of AxB

    AxB = {(a,1), (a, 2), (a, 3)}

    Possible subsets of AxB:

    C = {(a,1)}
    D = {(a,2)}
    E = {(a,3)}
    F = {(a,1),(a,2)}
    G = {(a,2),(a,3)}
    H = {(a,1),(a,3)}
    I = {(a,1), (a, 2), (a, 3)}
    J = {empty set}

    So all the subsets I listed above, C through J are all the possible relations between the sets A and B
     
  14. Feb 12, 2006 #13

    Hurkyl

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    {empty set} isn't a subset of AxB. (Because that is the set which contains the empty set, and is not the empty set itself) Writing {} for the empty set is common.
     
  15. Feb 12, 2006 #14

    matt grime

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    Hey, don't blame me for that perfectly accurate observation of what you were saying. You said that a relation was the same as a subset. Then you asserted that AxB has only one relation, that is you asserted it has exactly one subset. I asked you a rhetorical question 'you think that AxB has exactly one subset?' to point out where your error was. It does not have one subset. And I explained that only the empty set has one subset. Did I not point out that your assertion that there was one subset was unreasonable?
     
  16. Feb 12, 2006 #15
    Well, i listed all the possible subsets of AxB = {(a,1), (a, 2), (a, 3)}

    C = {(a,1)}
    D = {(a,2)}
    E = {(a,3)}
    F = {(a,1),(a,2)}
    G = {(a,2),(a,3)}
    H = {(a,1),(a,3)}
    I = {(a,1), (a, 2), (a, 3)}

    I can't see any one that I missed, unless its something in the definiton that i missed...(also, the order dosn't matter when listing sets, that that won't make a difference.)

    Just to confirm, {empty set} is NEVER a subset of any cartestian product right.
     
  17. Feb 12, 2006 #16

    Hurkyl

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    I think you misunderstood me.

    {empty set} is not a subset of AxB

    The empty set is.
     
  18. Feb 12, 2006 #17
    Yes looking back now i see what you were saying. I thought you were trying to say that I was way off... but I see my mistake now. :frown:
     
  19. Feb 12, 2006 #18

    Ohh! I see.
     
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