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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 $’0’, ‘2’, ‘5’$ and '$E$', are shown below.

Consider

1. push button pressed/not pressed is equivalent to logic $1 / 0$ respectively,
2. a segment glowing / not glowing in the display is equivalent to logic $1 / 0$ respectively.

If segments $a$ to $\mathrm{g}$ are considered as functions of $P_{I}$ and $P_{2}$, then which of the following is correct ?

1. $g=\bar{P}_{1}+P_{2}, \quad d=c+e$
2. $g=P_{1}+P_{2}, \quad d=c+e$
3. $g=\bar{P}_{1}+P_{2}, \quad e=b+c$
4. $g=P_{1}+P_{2}, \quad e=b+c$