Discussion
Which of the following is not correct ?
Answer: D
The incorrect relation is \(1 + \Delta = (E + 1)\nabla^{-1}\).
With \(Ef(x) = f(x + h)\), \(\Delta = E - 1\) and \(\nabla = 1 - E^{-1}\):
- \(E = 1 + \Delta\): true by definition.
- \(1 - \nabla = E^{-1}\), so \(\nabla(1 - \nabla)^{-1} = \nabla E = E - 1 = \Delta\): true.
- \(E^{-1}\Delta = E^{-1}(E - 1) = 1 - E^{-1} = \nabla\): true.
But \(\nabla^{-1} = E(E - 1)^{-1}\), so \((E + 1)\nabla^{-1} = E(E + 1)/(E - 1)\), which is not \(1 + \Delta = E\). So option D is the false one.
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