Consider the given figure. Suppose Δ U_1 and Δ U_2 represent the...

From the given P-V diagram, the process $A$ starts from an initial state (let us call it state 1) and ends at a final state (state 2). The process $B$ starts from state 2 and returns to state 1.

The change in internal energy for process $A$ is $\Delta U_1 = U_2 - U_1$.

The change in internal energy for process $B$ is $\Delta U_2 = U_1 - U_2$.

Adding the two changes in internal energy, we get:

$\Delta U_1 + \Delta U_2 = (U_2 - U_1) + (U_1 - U_2) = 0$

The combined process $A + B$ is a cyclic process, so the net change in internal energy is zero ($\Delta U = 0$).

According to the first law of thermodynamics for the combined process:

$\Delta Q = \Delta U + \Delta W$

Since $\Delta U = 0$, we have:

$\Delta Q = \Delta W \Rightarrow \Delta Q - \Delta W = 0$