Digital Logic Gates (Page 2 of 2) 

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Logic gates may be thought of as a combination of switches. For instance, the AND gate, whose output can only be '1' if all its inputs are '1', may be represented by switches connected in series, with each switch representing an input.  All the switches need to be activated and conducting (equivalent to all the inputs of the AND gate being at logic '1'), for current to flow through the circuit load (equivalent to the output of the AND gate being at logic '1'). 

An OR gate, on the other hand, may be represented by switches connected in parallel, since only one of these parallel switches need to turn on in order to energize the circuit load.

In Boolean Algebra, the AND operation is represented by multiplication, since the only way that the result of multiplication of a combination of 1's and 0's  will be equal to '1' is if all its inputs are equal to '1'.  A single '0' among the multipliers will result in a product that's equal to '0'.  The Boolean expression for 'A AND B' is similar to the expression commonly used for multiplication, i.e., A·B.

The OR operation, on the other hand, is represented by addition in Booelean Algebra. This is because the only way to make the result of the addition operation equal to '0' is to make all the inputs equal to '0', which basically describes an 'OR' operation.  The Boolean expression for 'A OR B' is therefore A+B.

The NOT operation is usually denoted by a line above the symbol or expression that is being negated:    A = NOT(A).  The NAND operation is simply an AND operation followed by a NOT operation.  The NOR operation is simply an OR operation followed by a NOT operation.  The symbols used for logic gates in electronic circuit diagrams are shown in Figure 1. 

Figure 1. Logic Gate Symbols

One of the most useful theorems used in Boolean Algebra is De Morgan's Theorem, which states how an AND operation can be converted into an OR operation, as long as a NOT operation is available.  De Morgan's Theorem is usually expressed in two equations as follows:

(A·B)  =  A  +  B; and

(A+B) =  A  ·  B.

De Morgan's Theorem has a practical implication in digital electronics - a designer may eliminate the need to add more IC's to the design unnecessarily, simply by substituting gates with the equivalent combination of other gates whenever possible.  Since NAND and NOR gates can be used as NOT gates, de Morgan's Theorem basically implies that any Boolean operation may be simulated with nothing but NAND or NOR gates.  This is why NAND and NOR gates are also called universal gates. 

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See Also:  RTL / DTL / TTL;  Boolean Algebra