In the truth tables, 1 symbolizes true, 0 is false. In the conjunctive AND operation, we see that only if both A and B are true is C true. In the disjunctive OR operation, if either A or B is true, then C is also true. From this seemingly naïve and obvious base, the entire Boolean system is built, and digital computers can perform not only complex mathematical operations, but logical ones as well, including the making of decisions on a purely logical basis.
Before going on to the few additional conditions and combinations that complete the algebra, let’s study some analogies that will make clear the AND/OR principles of operation. We can think of AND as two bridges in sequence over two rivers. We can reach our destination only if both bridges are working. However, suppose there are two parallel bridges and only one river. We can then cross if either or both of the bridges is working. A closer example is that of electrical switches. Current will flow through our AND circuit if—and only if—both switches are closed. When the switches are in parallel—an OR circuit—current will flow if either, or both, are closed.
The truth tables resemble the bridge or switch arrangements. We can proceed across the line of 1’s and 0’s in the first table only if both switches are closed. The symbol 1 means that the switch is closed, so we can cross only the bottom line. In the second table, we are told we can proceed across the line if either switch is closed. Thus we can cross lines 2, 3, and 4. We can use many symbols in our two-valued system.
Symbol
| Bridge | No Bridge |
| Power | No Power |
| 1 | 0 |
| True | False |
A little imagination suggests a logic computer of sorts with one switch, a battery, and a light bulb. Suppose we turn on the switch when we drive into our garage. A light in the hallway then indicates that the car is available. By using two switches we can indicate that a second car is also in the garage; or that either of them is, simply by choosing between AND logic and OR logic. Childish as this seems, it is the principle of even our most complex thinking processes. You will remember that the brain is considered a digital computer, since neurons can only be on or off. All it takes is 10 billion neuron switches!
Remington Rand UNIVAC
AND and OR gates in series. Switches 1 and 2, plus 3 or 4, are needed to light the bulb.
In addition to the conjunctives AND and OR, Boolean algebra makes use of the principle of negation. This is graphically illustrated thus:
| Original | Negation |
| A | Ā |
| 1 | 0 |
| 0 | 1 |