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Define a binary operation ** on the set ...

Define a binary operation `**` on the set `{0, 1, 2, 3, 4, 5}` as `a**b={a+b if a+b<6;` `a+b-6,if a+b geq 6`. Show that zero is the identity for this operation and each element `a !=0`of the set is invertible with `6-a` being the inverse of `a`

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Given `X={0,1,2,3,4,5}`
`a**b={a+b,if a+b<6;` `a+b-6,if a+b geq6`
To check if zero is the identity , we see that `a**0=a+0=a`
for all `a in x ` and also `0**a=0+a=a` for `a in x`
Given `a in x ,a+0<6 ` and also `0+a<6`
0 is the identity element for the given operation .
Now
The element `a in x` is invertible if there exists `b in x` such that `a xx b=e=b**a`
In this case , e=0 to `a**b=0=b**a`
`implies a**b ={a+b=0=b+a,if a+b<6;a+b-6=0=b+a-6 if a+b geq 6}`
`i.e. a=-b or b=6-a`
but since , `ab in x ={0,1,2,3,4,5}`
Hence `b=6-a` is the inverse of `a` i.e.`6-a`
`AA a in {1,2,3,4,5}`
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