We’re not going to worry too much about how light bulbs work. You only need to know this: in order for a light bulb to light up, an electric current has to run through the little wire call the filament.
And in order for that to happen, you need two thing: a source of power, such as a battery, and a closed circuit. Think about it this way: the close circuit allows electrons to flow through the bulb and light it up.
But if there’s a break in the circuit, no light.
Think of the gate on the top of the circuit like a light switch, sort of like the switches in a house. The only difference is that when you flip a switch in your house up, that means ON.
For our gate here, when you flip it DOWN, that closes the circuit and turns the light ON. If you flip it UP, there’s a break in the circuit; no electricity flows, and the light is OFF.
Gate down/closed = ON = True = 1
A simple switch like like this allows us to represent atomic sentences. Each atomic sentence has two states, true and false, and this circuit has two states: gate closed = bulb on vs. gate open = bulb off.
But how do we represent complex sentences? A complex sentence usually has multiple atomics, which we need to put on the same circuit in order for them to be part of the same complex sentence.
Each atomic sentence is one switch, which form the inputs. The light bulb, then, is the output.
Switch up/open = Atomic False
Bulb ON = Complex is True
Bulb OFF = Complex is False
Here’s how it works: when the P-switch is closed/down, that represents P is true. Ditto for Q.
And when this bulb is on, that means the complex sentence is true.
Now see if you can figure out this question:
Don’t worry if you find that problem hard. Once you see how it works it will make more sense.
This circuit has four states, just like a complex sentence with two atomic sentences. For example:
- Both P and Q are closed
- P is closed, Q is open
- P is open, Q is closed
- Both P and Q are open
Each state of the circuit represents a different row of the truth table.
If that makes sense, see if you can figure out which row of the truth table is pictured above.
The only way to turn the light on, that is to say, the only way to make the complex sentence true, is to close both gates. Put another way: the complex sentence is true just in case both inputs are true.
That is why this circuit is P&Q.
Next, try this problem.
Whenever you have gates on the same piece of wire like this, one after the other, it is called being in series.
Gates in series always means conjunction, because they have to all be closed in order for electricity to flow.
Now that you’ve got the basics, it’s time to see how to represent all of BOOL in circuits.