The only statement that holds for the given operation is Statement II
Operations and numbers
Given the following operartions
a [] b = a+b - ab/2
We need to check the true statement
For the expression x [] (y+z) = x [] y + x [] z
x [] (y+z) = x+y+z - x(y+z)/2
x [] (y+z) = x+y+z - (xy+xz)/2
x [] y +x [] z = x+y - xy/2 +[x+z - xz/2 ]
x [] y +x [] z = x+y+x+z -xy/2 - xz/2
x [] y +x [] z = 2x+y+z - (xy+xz)/2
This shows that the statement I is incorrect
For the second statement
y [] z = y+z - yz/2
x [] (y [] z) = x + (y+z - yz/2) - x(y+z-yz/2)/2
x [] (y [] z) = x+y+z-yz/2 -xy/2 - xz/2+xyz/4
For the other expression
x [] y = x+y - xy/2
(x [] y) [] z = x+y - xy/2 + z - z(x+y - xy/2)/2
(x [] y) [] z = x+y+z- xy/2 -zx/2 - zy/2 + xyz/4
This shows that x [] (y [] z) = (x [] y) [] z is correct (Statement II)
For the third statement
x [] z = x+z - xz/2
y [] z = y+z - yz/2
z [] 0 = z+0 - z(0)/2
z [] 0 = z
x[]z + y[]z - z[]0 = x+z - xz/2 + y+z - yz/2 - z
x[]z + y[]z - z[]0 = x+y+z - xz/2 - yz/2
For the expression (x+y)[] z
(x+y)[] z = (x+y)+z - (xyz)/2
Hence the statement III is not valid
Based on the explanations above, the only statement that holds for the given operation is Statement II
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