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    I have a question in linear algebra that uses XπY (X intersect Y)

    It is to do with subspaces X,Y,Z of a vector space V...e.g.

    (XπY)+(XπZ)=Xπ(Y+Z)...and you have to prove it or provide a counterexample.

    But, I dont really understand how intersect applys here...anyone care to enlighten me? Does it mean something is in this subset and that subset or something else ...cheers for clarifying this for me
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    I am in my 1st year...so this may be wrong.

    I believe the question is asking prove that the number of items in X and Y added to the number of items in X and Z is equal to the number of items in X and in (Y or Z), or in other terms, is the intersect an associative operation under these conditions,ie can you carry out the sum total of the number of members of the 2 subsets THEN take the intersect of them, instead of just doing the intersects seperatly and adding the number of members in the 2 intersections.

    :confused: . Ive confuzled myself now .

    Anyway, I dont think I know enough maths to solve it, so the rest is up to you and the rest of the people
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    I don't think the question will be asking about 'numbers of elements in, say, XπY, but rather the actual elements they contain.

    I think it would help to have what '+' means in the case of X+Y etc.

    If it is what I think it should mean, then X+Y is the set of all x+y such that x is in X and y is in Y. The same applies for the 'sets' we have in the question.

    As for intersection, XπY is the subset of all the v in V such that each v is both in X and in Y.

    As for proving the question, the best way may be to pick a general element out of the right hand side and prove it is also in the left hand side. This shows the right hand side is a subset of the left hand side.

    The you should take a general element of the left hand side and prove it is also in the right hand side. This proves the left hand side is a subset of the right hand side.

    Since both are subsets of each other, then the two sets must be equal.

    Hope this helpes
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    Oops...forgot that method...

    /me reminds himself to listen to his lecturers in future...
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    (Original post by Hitonagashi)
    Oops...forgot that method...

    /me reminds himself to listen to his lecturers in future...
    What? You forgot the method of showing each side of the equals is a subset of the other?

    Don't worry, I never used to think of doing this for ages....took me quite a few wrong homework questions for me to automatically think of doing this when I see a question...doesn't always work, but often does (I hope it does in this case). I'm sure you'll get better at spotting the right approach to hit a question with as your course continues
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    (Original post by lgs98jonee)
    I have a question in linear algebra that uses XπY (X intersect Y)

    It is to do with subspaces X,Y,Z of a vector space V...e.g.

    (XπY)+(XπZ)=Xπ(Y+Z)...and you have to prove it or provide a counterexample.

    But, I dont really understand how intersect applys here...anyone care to enlighten me? Does it mean something is in this subset and that subset or something else ...cheers for clarifying this for me
    If you take (as subspaces of R^2)

    X to be the x-axis
    Y to be the y-axis
    Z to be the y=x line

    the X n Y = X n Z = 0 and so their sum is 0

    But Y+Z = R^2 and so the RHS is X.
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    (Original post by Neapolitan)
    If you take (as subspaces of R^2)

    X to be the x-axis
    Y to be the y-axis
    Z to be the y=x line

    the X n Y = X n Z = 0 and so their sum is 0

    But Y+Z = R^2 and so the RHS is X.
    Or just forget what I said about how to prove the statement is true as no matter how good a method you use, if it ain't true it ain't going to work....to be honest I can't actually remember reading anything about giving a counter example in the original post...I should read things more closely in future ....but anyway, this is a good counter example
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    Does X+Y=X U Y? I still dont understand the difference between these two
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    If V = X+Y then if v \in V, x \in X and y \in Y you can write v = x+y.
    As Neapolitan mentions, if you can X the x axis and Y the y axis, then V is the whole R^{2} plane because you can move so many along and so many up/down to get to somewhere in the plane.

    If V = X \cup Y then V is any point on the X axis and any point on the Y axis. The point (1,1) isn't on the X or Y axis, so isn't in V.
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    X+Y is the smallest subspace containing X and Y

    XUY is the smallest subset containing X and Y
 
 
 
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