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Consider the following binary operations on Z. which of the following operations form a group
i. x*y = x+y+2   for x,y ∈ Z.
ii. x*y = xy + 2x + 2y +2 for x,y ∈ Z
iii. x*y=x2y2 for x,y ∈ Z

asked in Set Theory & Algebra by Veteran (14.4k points)   | 56 views

1 Answer

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i. * closed operation, x*y belongs to Z
ii. Associative
iii.    x*e = x
        x + e + 2 = x
        e = -2      // identity element exists
iv.  x*y=e   
      x+y+2 = -2
       x+y=-4
           y = (-4 -x)    inverse exist
v. Commutative

Hence statement 1 is Abelian Group!

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Statement 2:  Not Associative

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Statement 3. 
x*y=x2yfor x,y ∈ Z   not associative not even semi group

answered by Veteran (14.4k points)  
edited by
thank you @sushant for the correction!
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