Given the equations P = _ Δ S → 0 (F)/(Δ S) and P_1 - P_2 = g z ,...

The first equation $P = \lim_{\Delta S \to 0} \dfrac{F}{\Delta S}$ represents the fundamental definition of pressure at a point. This definition is independent of the frame of reference and remains valid in an accelerating elevator.

The second equation $P_1 - P_2 = \rho g z$ represents the hydrostatic pressure difference in a fluid at rest in an inertial frame. In an elevator accelerating upward with an acceleration $a$, the effective acceleration due to gravity becomes $g_{eff} = g + a$. The pressure difference in this non-inertial frame is given by $P_1 - P_2 = \rho (g + a) z$. Thus, the second equation is not valid in its given form.

Therefore, the first equation is valid but the second is not.