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9.4 BOOLean Logic Gates

All of these circuits that represent logical formulas are called logic gates.

You’ve already learned how to represent conjunction with electrical circuits. All of these circuits that represent logical formulas are called logic gates.

Next, see if you can figure out this problem.

Notice how this circuit is different from the gates that we saw in series before: the P and Q gates are now on different pieces of wire.

That means that if just one of them is closed, the circuit is closed and the bulb will light up.

PvQ circuit with P closed and Q open. Light on.

Now let’s see if you understand the diagram:

Parallel gates: on independent pieces of wire, so that one can affect the circuit without the other.
Parallel gates = disjunction.

When gates are separated on different pieces of wire, they are called parallel. Parallel gates are represented by disjunctions, since if either one is closed, the light bulb will go on.

Also note this: in our example, the two gates are stacked and geometrically parallel to each other, but whether the gates count as parallel isn’t exactly determined by the geometry. We could have put one of the wires and gates off kilter and at an angle, and they would still be parallel in the logical sense.

Here is more practice.

Now all we have to do is represent negation, and we’ll have a complete set of Boolean logic gates.

Negation inverts the truth value of a sentence. In circuits, it should invert the value of the current: when the gate is closed/on, the bulb should be off.

Negation: moving the contact so “open” position completes the circuit.

Consider the lights in your house: flipping the switch UP turns it on and DOWN turns it off. Negation would mean that switch UP/ON = light OFF and switch DOWN/OFF = light ON.

How could you do that? Just use a screwdriver to take the switch out and turn it upside down!

What that does, from a physical point of view, is change where the contact points are. Now when the switch is DOWN/OFF, that completes the circuit, which is why the light goes on.

To create this in our diagrams, all we have to do is move the terminal that the end of the gate hits, so that when the gate is OPEN/OFF, it completes the circuit:

Not-P circuit with P off. Light on.

Whenever a gate is represented at an upward diagonal, that means it is false. Whenever a gate is horizontal, that means it is true.

The key to understanding this diagram is to realize that P is in the FALSE position (it is OPEN/OFF), even though what that does is “close” the circuit. And now when you make P TRUE, the bulb goes off.

Not-P circuit with P on. Light off.

If you understand that, see if you can figure out this problem.

Now that you know the logic gates for all the Boolean operators, with a little practice you’ll be able to figure out the Boolean form of any electrical circuit.