The "mathematics of logic," developed by English mathematician George Boole in the mid-19th century. Its rules govern logical functions (true/false) and are the foundation of all electronic circuits in the computer. As add, subtract, multiply and divide are the primary operations of arithmetic, AND, OR and NOT are the primary operations of Boolean logic. Boolean logic is turned into logic gates on the chip, and the logic gates make up logic circuits that perform functions such as how to add two numbers together.
Various permutations of AND, OR and NOT are used, including NAND, NOR, XOR and XNOR. The rules, or truth tables, for AND, OR and NOT follow. See
Boolean search,
binary,
logic gate and
Bebop to the Boolean Boogie.
Curious About the Chip?
Wired in patterns of Boolean logic and in less space than a postage stamp, transistors inside one of today's high-speed chips collectively open and close trillions of times every second. If you're curious about how it really works down deep in the layers of the silicon, read the rest of "Boolean logic," then "chip" and, finally, "transistor." It's a fascinating venture into a microscopic world.
The logic of AND, OR and NOT is implemented as transistors, which are electronic switches that are opened and closed by being pulsed. If you don't understand every last detail below, keep on going. It will come together at the end. You can always review.
AND
AND requires both inputs to be present in order to provide output. Because the AND gate is wired in series, both inputs must pulse both switches closed, and current flows from the source to the output.
OR
OR requires one input to be present in order to provide output. Because the OR gate is wired in parallel, either one or both inputs will generate output.
NOT
NOT reverses the input. If there is no pulse on the input line, the source goes directly to output, as in the diagram above. If there is a pulse on the input line, switch #1 is closed. The switch #1 current goes to switch #2 and pulses it open (it's a reverse switch), and the source current is impeded.
The Hierarchy
The gates make up circuits, and circuits make up logical devices, such as a CPU. We're going to look at a circuit that is present in every computer. It adds one bit to another.
Adding Two Bits Together
The half-adder circuit adds one bit to another and yields a one-bit result with one carry bit. This circuit in combination with a shift register, which moves over to the next bit, is how a string of binary numbers are added. This diagram shows the four possible binary additions for two bits.
The Half-Adder Circuit
Trace the current through the example above. See how AND, OR and NOT react to their inputs. The 1 is represented in red (flow of current), and the 0 in blue (no current). Try it yourself below.
Try It Yourself
Print this diagram and try your Boolean skill. Review the combinations of 0 and 1 above and pick any pair. With a pen or pencil, draw a line to represent a 1. Draw nothing for 0, and see if you can get the right answer.
