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For (R,o), where xoy=x+y+sqrt3 deretmine wheter or not it forms a group, by checking...

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plinkster | Student, College Freshman | eNotes Newbie

Posted March 30, 2012 at 5:38 AM via web

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For (R,o), where xoy=x+y+sqrt3 deretmine wheter or not it forms a group, by checking the group axioms.

Can somebody just check it for me please. Not really sure if I`m right in that what i have done (:

I found that:
G1- xoy belogns to R, for any x,y in R  so xoy is closed under o .
G2. x o e:  x+e+sqrt3=x that is e=-sqrt 3
      e o x: e+x+sqrt3=x
, so -sqrt3 is an identity. --- not relly sure if that correct.

G3. x o y= e  so x+y+sqrt3=-sqrt3
                    thus inverse of x = y= -x-2sqrt3
                    and g o g^(-1)=e=g(-1) o g

G4 LHS= x o(yoz)= xo(y+z+sqrt3)=z+y+z+2sqrt3
     RHS-(x o y)o z= (x+y+sqrt3)oz=x+y+z+2sqrt3
So LHS=RHS

Hence (R,o) is a gorup.

 

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted March 31, 2012 at 2:00 AM (Answer #1)

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You need to check if the 4 axioms hold such that:

1. Closure axiom: For all `g_1, g_2 in R =gt g_1og_2 in R`

Hence, for all `x,y in R`  yields `xoy=x + y + sqrt3 ` `in R`  (the addition of three real numbers, x+y+sqrt3, is also a real number)

2. Identity axiom: There is an identity element `e in R`  such that:

`xoe = eox = x`

`xoe = x+e+sqrt3`

You need to set the equations `x+e+sqrt3`  and x equals such that:

`x+e+sqrt3 = x =gt e+sqrt3 = 0 = gt e = -sqrt 3`

3. Inverses axiom: For each `x in R` , there is an inverse `x^(-1)``in R`  such that:

`x o x^(-1) = x^(-1) o x = e`

`x o x^(-1) = x + x^(-1) + sqrt 3`

`x + x^(-1) + sqrt 3 = -sqrt 3 =gt x^(-1) = -x - 2sqrt3`

4. Associativity axiom: For all `x,y,z in R`  yields:

`xo(yoz) = (xoy)oz`

`x + (yoz) + sqrt 3 = (xoy) + z + sqrt 3`

Reducing `sqrt 3`  both sides yields:

`x + (yoz) = (xoy) + z`

You need to substitute `y+z+sqrt3`  for (yoz) and `x+y+sqrt3 ` for (xoy) such that:

`x + y + z + sqrt3 = x + y + sqrt3 + z`

Notice that both sides are equal.

Hence, since the 4 axioms hold, then the set R forms a group with binary operation "o".

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