Two products are sold from a vending machine, which has two push buttons $P_{1}$ and $P_{2}$. When a button is pressed, the price of the corresponding product is displayed in a $7$-segment display.
- If no buttons are pressed, $’0’$ is displayed, signifying 'Rs. $0$'.
- If only $P_{1}$ is pressed, $’2’$ is displayed, signifying 'Rs. $2$'.
- If only $P_{2}$ is pressed, $’5’$ is displayed, signifying 'Rs. $5$'.
- If both $P_{1}$ and $P_{2}$ are pressed, ' $\mathrm{E}$ ' is displayed, signifying 'Error'.
The names of the segments in the $7$-segment display, and the glow of the display for $\text{‘0’, ‘2’, ‘5’}$ and $\text{‘E’},$ are shown below.
Consider
- push button pressed/not pressed is equivalent to logic $1 / 0$ respectively,
- a segment glowing / not glowing in the display is equivalent to logic $1 / 0$ respectively.
If segments $a$ to $g$ are considered as functions of $P_{I}$ and $P_{2}$, then which of the following is correct?
- $g=\overline{P}_{1}+P_{2}, \quad d=c+e$
- $g=P_{1}+P_{2}, \quad d=c+e$
- $g=\overline{P}_{1}+P_{2}, \quad e=b+c$
- $g=P_{1}+P_{2}, \quad e=b+c$