Question

Let [] denote the operation a [] b = a+b - ac{ab}{2}" alt="\frac{ab}{2}" align="absmiddle" class="latex-formula"> ....

Answer 1

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

Learn more on operations and numbers here: brainly.com/question/2292256

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