expression: , a AND b = y

truth table:

expression: a + b = y, a OR b = y

truth table:

expression: , NOT a = y

truth table:

expression: , a NAND b = y

truth table:

expression: , a NOR b = y

truth table:

expression: , a XOR b = y

truth table:

Truth table based design is probably the easiest and most efficient approach in designing a logic circuit. The use of a truth table, as the basis for the design of a circuit, permits the use of Boolean algebra. (Boolean algebra for logic elements can be learned here.) Therefore, this approach is highly objective and less dependent on the designer's skill and experience.

All you have to do to put your idea in truth table form is recognize all inputs and outputs of your circuit and give each one an individual variable name. Figure out what combinations of input-states (1s or 0s) would be valid for the use of your circuit. Then organize all inputs and outputs in a table like this example:

If the majority of possible output signals for a given output are 0s, then you should use the sum of products method.

In the above example, the sum of products method would be used. We first take the 1s in the given output (y) and focus on these entire horizontal segments of binary digits. In sum of products, we need to turn all the inputs to 1. We do this by "NOTing" each of the 0 inputs. We must also "AND" each individual input together. We do the same to the lines containing the other 1 outputs and then we OR these two expressions together.

The product of sums method is the opposite of the sum of products method.

With this method, you focus on the outputs that are 0s. In the same manner as the sum of products, you focus on the lines containing 0 outputs and this time change all of the 1 inputs to 0s by "NOTing" them. Instead of "ANDing" each of these inputs together, you will "OR" them together. Do this to all other lines with 0s as outputs and finally "AND" them all together.

Although this statement looks differently than the one obtained by the sum of products method, the have the exact equivalence and can be determined by "Boolean Expression Simplifications", which can be found here.

Boolean expression simplifications can be directly applied to any Boolean expression. What these simplifications do is reduce the number of digital elements needed to design a circuit but retain its normal functions. Below is a list of some basic simplifications for digital circuits using AND, OR, NOT, NAND, NOR, and XOR elements.