Switches, Lights, and Logic! Venus has switches at the top and bottom of her stairs to control the light for the stairwells. (Why she has lights on the side of her stairs, I don't know...) She notices that when the upstairs switch is up, and the downstairs switch is down, the light is turned on. **Questions:** $$\\\$$ **(a)** Copy and complete the truth table below: ** If both the upstairs and downstairs switches are in the up position, will the light be on? Explain your reasoning. **(c)** If the upstairs switch is in the down position, and the downstairs switch is in the up position, will the light be on? Explain. **(d)** In general, how should the two switches be positioned such that the light is on? I'm confused as to where to begin. Even a little hint is nice too.

Each switch can only be in 2 possible positions, up or down. So all possible states:

$$ \begin{matrix} Upstairs & Downstairs & Light On \\\ Up & Up & \\\ Up & Down & True \\\ Down & Up & \\\ Down & Down & \end{matrix} $$

Notice that the light is on if the switches are in different positions. If you "toggle" one of the lights from Up/Down to, say, Up/Up, that will turn the light off. Toggle like this until you have your complete table, and the rest of the answers will come directly from that table.

(This is basically just an exclusive-or)