@: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.
- Anonymous1 decade agoFavorite Answer
(1) a * b =( 1/5)(a+1)(b+1)
(2) The identity for * on Q is 4.
(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
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.
In order to complete the proof, you should check yourself that
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.
- 1 decade ago