For this type of questions you need to manipulate the variables and try to reduce the given function to a boolean expression which is functionally complete.
Absorption Law: a + a’b = a + b (for reference)
Consider option A,
f(x,y,z) = x’ + yz’
Now, substitute y in place of x in the above function i.e.
f(y,y,z) = y’ + yz’ = y’ + z’ = (y.z)’
which is nothing but a nand gate and we know that it is universal gate and hence it is functionally complete. Since the given expression can be reduced down to behave as a nand gate we can implement any other function with it.
option C,
f(x,y,z) = xy’ + x’ + x’z = xy’ + x’(1 + z) = xy’ + x’ = x’ + y’ = (x.y)’ (again a nand gate)
Option B,
f(x,y) = x’ + xy = x’ + y
this is not a standard universal function. Hence not functionally complete.
Hence, correct options (A) & (C).
