The only statement that holds for the given operation is Statement II
<h3>Operations and numbers</h3>
Given the following operartions
a [] b = a+b - ab/2
We need to<u> check the true statement</u>
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
Learn more on operations and numbers here: brainly.com/question/2292256
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