1 of 2

9.3 Electricity

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 called the filament.

And in order for that to happen, you need two things: a source of power, such as a battery, and a closed circuit. Think about it this way: the closed circuit allows electrons to flow through the bulb and light it up.

Closed circuit with light bulb on.

But if there’s a break in the circuit, no light.

Open circuit with light bulb off.

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.

Atomic sentence:
Gate down/closed = ON = True = 1

A simple switch 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.

For example:

Open circuit with P and Q gates open. Light bulb off.

Switch down/closed = Atomic True
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.

Each state of the circuit represents a different row of the truth table

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.

Closed circuit with P and Q gates closed. Light bulb on.

That is why this circuit is P&Q.

Next, try this problem.

Another name for switches is gates.

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.

Gates in series = Conjunction, &.

Now that you’ve got the basics, it’s time to see how to represent all of BOOL in circuits.