Digital Logic and Basic Functions and gates ,
(logic gates)
About Digital Logic
Digital or Binary logic has fascinated many people over the years. The very idea that a two-valued number system can possible be the basic for the most powerful and sophisticated computers seem- astounding, to say the least. Nevertheless, it is so and the how and the why of this requires some Explanation. Everything in the digital world is based on the binary number system. Numerically, this involves only two symbols: 0 and 1 . Logically, we can use these symbols or we can equate them with others according to the needs of the moment. thus, when dealing with digital logic, we can specify that:
0 = false = no
1 = true = yes
Using this two-valued logic system, every statement or condition must be either "true" , it cannot be partly true and partly false. While this approach may seem may seem limited, it actually works quit nicely, and can be expanded to express very complex relationships and interactions among any number of individual conditions. One essential reason for basing logical operations on the binary number system is that it is easy to design simple, stable electronic circuits that can switch back and forth between two clearly-defined states, with no ambiguity attached. it is also readily possible to design and build circuits that will remain indefinitely in one state unless and until they are deliberately switched to the other state. this makes it possible to construct a machine (the computer) which can remember sequences of events and adjust its behavior accordingly.
Digital logic may be divided into two classes: combinational logic, in which logic outputs are determined by the logical function being performed and the logical input states at that particular moment and sequential logic, in which the outputs also depend on the prior states of those outputs. both classes of logic are used extensively in all digital computers. since both types of logic circuits begin with logic gates to combine logical input signals in various way to produce the desired outputs, we will begin on the next page by seeing how the basic logic gates work.
Using this two-valued logic system, every statement or condition must be either "true" , it cannot be partly true and partly false. While this approach may seem may seem limited, it actually works quit nicely, and can be expanded to express very complex relationships and interactions among any number of individual conditions. One essential reason for basing logical operations on the binary number system is that it is easy to design simple, stable electronic circuits that can switch back and forth between two clearly-defined states, with no ambiguity attached. it is also readily possible to design and build circuits that will remain indefinitely in one state unless and until they are deliberately switched to the other state. this makes it possible to construct a machine (the computer) which can remember sequences of events and adjust its behavior accordingly.
Digital logic may be divided into two classes: combinational logic, in which logic outputs are determined by the logical function being performed and the logical input states at that particular moment and sequential logic, in which the outputs also depend on the prior states of those outputs. both classes of logic are used extensively in all digital computers. since both types of logic circuits begin with logic gates to combine logical input signals in various way to produce the desired outputs, we will begin on the next page by seeing how the basic logic gates work.
Basic Logical Functions and Gates
While each logical element or condition must always have a logic value of either "0" or "1" , we also need to have ways to combine different logical signals or conditions to provide a logical result.
For example, consider the logical statement: "If move the switch on the wall up, the light will turn on." At first glance, this seems to be a correct statement. However, we look at a few other factors, we realize that there's more to it than this. In this example, a more complete statement would be: "If move the switch on the wall up and the light bulb is good and the power is on, the light will turn on,"
If we look at these two statements as logical expressions and use logical terminology, we can reduce the first statement to:
Light = switch
This means nothing more than that the light will follow the action of the switch, so that when the switch is up/on/true/ 1 the light will also be on/true/1. conversely, if the switch is down/off7false/0 the light will also be off/false/0.
Light = Switch and bulb and power
Normally, We use symbols rather than words to designate the and function that we' re using to combine the separate variables of Switch, bulb, and Power in this expression. The symbol normally used is a dot, Which is the same symbol used for multiplication in some mathematical expressions. Using this symbol, our three-variable expression becomes:
Light = Switch • Bulb • Power
When We deal with logical circuits (as in computers), We not only need to deal with logical functions; We also need some special symbols to denote these functions in a logical diagram. There are three fundamental logical operations, from Which all other functions, no matter how complex, can be derived. These functions are named and, or, and not. Each of these has a specific symbol and a clearly-defined behavior, as follows:
The AND Gate
The AND gate implements the AND Function. the gate shown to the left, both inputs must have logic 1 signals applied to them in order for the output to be a logic 1. With either input at logic 0, the output will be held to logic o.
There is no limit to the number of inputs that may be applied to an AND function, so there is no functional limit to the number of inputs an AND gate may have. However , for practical reasons, commercial AND gates are most commonly manufactured with 2,3, or 4 inputs. A standard integrated Circuit (IC) package contains 14 or 16 pins, for practical size and handling. A standard 14-pin package can contain four 2-input gate, three 3-input gates, or two 4-input gates, and still have room for two pins for power supply connections
AND Operation
The expression X = A * B read as "X equals A AND B ".
The multiplication sign stands for the AND operation, same for ordinary multiplication of 1s and 0s.
The AND gate operation produces a result of 1 occurs only for the single case when all of the input variables are 1.
The output is 0 for any case where one or more inputs are 0.
Two inputs AND gate and its table is bellow :
Three inputs AND gate and its truth table is bellow :
Normally, We use symbols rather than words to designate the and function that we' re using to combine the separate variables of Switch, bulb, and Power in this expression. The symbol normally used is a dot, Which is the same symbol used for multiplication in some mathematical expressions. Using this symbol, our three-variable expression becomes:
Light = Switch • Bulb • Power
When We deal with logical circuits (as in computers), We not only need to deal with logical functions; We also need some special symbols to denote these functions in a logical diagram. There are three fundamental logical operations, from Which all other functions, no matter how complex, can be derived. These functions are named and, or, and not. Each of these has a specific symbol and a clearly-defined behavior, as follows:
The AND Gate
The AND gate implements the AND Function. the gate shown to the left, both inputs must have logic 1 signals applied to them in order for the output to be a logic 1. With either input at logic 0, the output will be held to logic o.
There is no limit to the number of inputs that may be applied to an AND function, so there is no functional limit to the number of inputs an AND gate may have. However , for practical reasons, commercial AND gates are most commonly manufactured with 2,3, or 4 inputs. A standard integrated Circuit (IC) package contains 14 or 16 pins, for practical size and handling. A standard 14-pin package can contain four 2-input gate, three 3-input gates, or two 4-input gates, and still have room for two pins for power supply connections
AND Operation
The expression X = A * B read as "X equals A AND B ".
The multiplication sign stands for the AND operation, same for ordinary multiplication of 1s and 0s.
The AND gate operation produces a result of 1 occurs only for the single case when all of the input variables are 1.
The output is 0 for any case where one or more inputs are 0.
Two inputs AND gate and its table is bellow :
 |
| (logic gates) |
 |
| (three inputs logic gate) |
The OR Gate
The OR gate is sort of the reverse of the AND gate. The OR function, like its verbal counterpart, allows the output to be true if any one or more of its inputs are true. verbally, we might say, "If it is raining OR if turn on the sprinkler, the lawn will be wet ." Note that the lawn will still be wet if the sprinkler is on and it is also training. this is correctly reflected by the basic OR function. in symbols, the OR function is designated with a plus sign (+). in logical diagrams, the symbol to the left designates the OR gate. As with the AND function, the OR function can have any number of inputs. However, practical commercial OR gates are mostly limited to 2,3, and 4 inputs, as with AND gates.
OR Operation
The expression X = A + B read as "X equals A OR B".
The + sign stands for the OR operation, not for ordinary addition.
The OR operation produces a result of 1 when any of the input variable is 1.
The OR operation produces a result of 0 only when all the input variables are 0.
two inputs OR gate and its truth table is as bellow :
 |
| (two input or gate) |
 |
| (three input or gate) |
The NOT gate or Inverter
The inverter is a little different from AND and OR gates in that it always has exactly one input as well as one output. whatever logical state is applied to the input, the opposite state will appear at the output. the NOT function, as it is called, is necessary in many applications and highly useful in others. A practical verbal application might be. the door is NOT locked = you may enter the NOT function is denoted by a horizontal bar over the value to be inverter, as shown in the figure to the left. in some case a single quote mark (') may also be used for this purpose: 0' = 1 and 1' = 0. for greater clearity in some logical expressions, we will use the overbar most of the time.
in the inverter symbol, the triangle actually denotes only an amplifier, which in digital term means that it "cleans up" pie signal but does not change its logical sense. it is the circle at the output which denotes the logical inversion. the circle could have been placed at the input instead, and the logical meaning would still be the same for example, if the variable A is subjected to the NOT operation, the result x can be expressed as x = A' where the prime(') represents the NOT operation. this expression is read as:
x equals NOT A
x equals the inverse of A
x equals the complement of A
Each of these is in common usage and all indicate that the logic value of x = A' is opposite to the logic value of A. The truth table of the NOT operation NOT operation is as follows.
NOT Operation
The truth table of the NOT operation NOT operation is as follows:
 |
| (not logic gate with transistor switch) |
1 = 0 Because NOT 1 is 0
0 = 1 Because NOT 0 is 1
The NOT operation is also referred to as inversion or complementation, and these term are used interchangeably.the logic gates shown above are used in various combinations to perform tasks of any level of complexity. some functions are so commonly used that they have been given symbols of their own and are often packaged so as to provide that specific function directly. on the next page, we'll begin our coverage of these functions.
Derived logical functions and gates
While the three basic functions AND, OR, and NOT are sufficient accomplish all possible logical functions and operations, some combinations are used so commonly that they have been given names and logic symbols of their own. We will discuss three of these on this topic. the first is called NAND, and consists of an AND function followed by a NOT function. The second, as you might expect, is called NOR. this is an OR function followed by NOT the third is a variation of the OR function, called the Exclusive-OR, or XOR function. As with the three basic logic functions. each of these derived functions has a specific logic symbol and behavior,which we can summarize as follow:
The NAND gate
The NAND gate implements the NAND function, which is exactly inverted from the AND function you already examined. with the gate shown to the left, both inputs must have logic 1 signals applied to them in order for the output to be a logic 0. with either input at logic 0, the output will be held to logic 1.
The circle at the output of the NAND gate denotes the logical inversion, just as it did at the output of the inverter. Also in thefigure to the left, note that the overbar is a solid bar over both input values at once. this shows that it is the AND function itself that is inveterd , rather than each separate input. NAND is the same as the AND gate symbol except that it has a small circle on the output. this small circle represents the inversion operation. therefore, the output expression of the two input NAND gate is:
x = (AB)
As with AND, there is no limit to the number of inputs that may be applied to a NAND function, so there is no functional limit to the number of inputs a NAND gate may have. However, for practical reason , commercial NAND gates are most commonly manufactured with 2,4 or 4 inputs, to fit in a 14-pin or 16-pin package.
|
| (nand logic gate) |
As with AND, there is no limit to the number of inputs that may be applied to a NAND function, so there is no functional limit to the number of inputs a NAND gate may have. However, for practical reason , commercial NAND gates are most commonly manufactured with 2,4 or 4 inputs, to fit in a 14-pin or 16-pin package.
The NOR gate
The NOR gate is an OR gate with the output inverted. where the OR gate allows the output to be true (logic 1) if any one or more of its inputs are true, the NOR gate inverts this and forces the output to logic 0 when any input is true. in symbols, the NOR function is designated with a plus sign (+), with an overbar over the entire expression to indicate the inversion. in logical diagrams, the symbol to the left designates the NOR gate. As expected, this is an OR gate with a circle to designate the inversion. NOR is the same as the OR gate symbol except that it has a small circle on the output. this small circle represents the inversion operation. therefore the output expression of the two input NOR gate is:
X = ( A + B)'
|
| (nor logic gate) |
The Exclusive-OR, or XOR Gate
The exclusive-OR, or XOR function is an interesting and useful variation on the basic OR function. Verbally, it can be stated as, "Either A or B, but not both." The XOR gate produces a logic 1 output only if its two inputs are different. if the inputs are the same, the output is a logic 0. The XOR symbol is a variation of the standard OR symbol . it consists of a plus (+) sign with a circle around it. the logic symbol, as shown here, is a variation on the standard OR symbol. Unlike standard OR/NOR and AND/NAND functions, the XOR function always has exactly two inputs, and commercially manufactured XOR gate are the same. four XOR gates fit in a standard 14-pm IC package the three derived functions shown above are by no means the only ones, but these from the basic of all the others. Next we will begin our look at practical applications for logic gates in various combinations, to see just how these simple gate can be combined to perform every possible operation in a computer.
- Choose the correct answers in the following questions.
- Boolean algebra is different from ordinary algebra in which way?
- Boolean algebra can represent more than 1 discrete level between 0 and 1
- Boolean algebra has only 2 discrete levels: 0 and 1
- Boolean algebra can describe up to 3 logic levels
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