青ㄟ
Lv 6
青ㄟ asked in 科學數學 · 1 decade ago

代數學-請高手回答

@:Q-->Q defined by @(x)=5x-1 for x€Q is one to one and onto Q.Give the definition of a binary operation * on Q such that @ is an isomorphism mapping <Q, .(一點)>onto<Q,*>and then give the identity element for * on Q.

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  • Anonymous
    1 decade ago
    Favorite Answer

    Answers:

    (1) a * b =( 1/5)(a+1)(b+1)

    (2) The identity for * on Q is 4.

    Solutions:

    (1)At the first, try to find the definition of the binary operation* on Q for

    which @ is an isomorphism.

    Since @ is an isomorphism, so @ (x.y) =@(x) * @(y) ,

    Hence, 5xy-1=(5x-1) *( 5y-1)----(#).

    Let a=5x-1, b=5y-1, then x=(1/5)(a+1), y=(1/5)(b+1).

    Replace x and y in terms of a and b in the equation(#),

    we got 5(1/5)(a+1)(1/5)(b+1)=a*b, therefore we should define

    a*b= (1/5)(a+1)(b+1)

    so that @ is an isomorphism.

    (2)Next, try to find the identity element for * on Q.

    If b is the identity , then a*b=a, for all a in <Q,*>.

    This leads to (1/5)(a+1)(b+1)=a, from this equation of b,

    we got b = 4.

    Therefore, the desired identity for * on Q is 4.

    (3)

    In order to complete the proof, you should check yourself that

    we defined

    a * b =( 1/5)(a+1)(b+1)

    in this way is indeed a binary operation on Q .

    Also, check that 4 is the desired identity for * on Q .

    Anyway, these are easy routine works.

  • 青ㄟ
    Lv 6
    1 decade ago

    b怎會等於4

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