Discussion
Let \(\Delta\) be the forward difference operator, \(\nabla\) be the backward difference operator and \(\delta\) be the central difference operator. Then :
Answer: A
The correct relation is \(\nabla \Delta = \Delta - \nabla\).
Write the operators with the shift E: \(\Delta = E - 1\), \(\nabla = 1 - E^{-1}\), \(\delta = E^{1/2} - E^{-1/2}\). Then
\[ \begin{aligned} \nabla\Delta &= (1 - E^{-1})(E - 1) = E - 2 + E^{-1} \\ \Delta - \nabla &= (E - 1) - (1 - E^{-1}) = E - 2 + E^{-1} \\ \delta^{2} &= E - 2 + E^{-1} \end{aligned} \]
So \(\nabla\Delta = \Delta - \nabla = \delta^{2}\). That makes option A true and options B, C and D, each of which asserts one of these equalities fails, false.
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