# Which of these statements is not true Solution: The answer will be an option, (a) If f(x) is (g(x)) then g(x) is (f(x)) Explanation: =>Option (a) is not true because using definition of big-omega if f(x) = (g(x)) then we can write f(x) C*g(x) where C is some constant, x x0, x0 1 so we can not write g(x) D*g(x) where D is some constant, x x0, x0 1. =>Option (b) is true because using definition of big-omega if f(x) = (g(x)) then we can write f(x) C*g(x) where C is some constant, x x0, x0 1 so we can write g(x) = O(f(x)) using definition of big-O as f(x) D*g(x) where D is some constant, x x0, x0 1 and both relations are asymptotically equivalent. =>Option (c) is true because using definition of big-theta we can write C*g(x) f(x) D*g(x) where C and D are constants, x x0, x0 1 hence we can write g(x) = O(f(x)) as it satisfies the definition of big-O as f(x) E*g(x) where E is some constant, x x0, x0 1. =>Option (d) is true because using definition of big-theta we can write C*g(x) f(x) D*g(x) where C and D are constants, x x0, x0 1 hence we can write g(x) = (f(x)) as definition of big-theta E*g(x) f(x) F*g(x) where E and F are constants, x x0, x0 1 hence both relations are equivalent. I have explained each and every part with the help of statements attached to the answer above.