**LOGIC AND SWITCHING FUNCTIONS**

A logic function is defined as

*a function that obeys the rules governing a logical statement*. A logical switching network may be defined*as a network formed by inter-connecting a finite number of switches so as to obey the logic rules associated with a logical statement.*Let*z*be a logic function. We may express this as*z = f*(

*x*

_{i})

*(1)*

where

*x*_{i}*= x*_{1},*x*_{2}, …,*x*_{m }represent the inputs to the switching network, and*z*is its output. Equation (1) can be treated as a logic function, when we define the inputs and their mutual logical relations to give the output*z*. It may be noted that*x*_{i}’s and*z*assume only two values, viz.,**0**and**1**.
A logic system can be tested for its performance by using what is called its

*truth**table*. In truth tables, entries are made using**0**s and**1**s. In practical circuit applications, a turned-on switch can be used to represent binary**0**and a turned-off switch can be used to represent binary**1**, or vice versa. Combinations of these switches can be used to generate any desired logic function. Such functions, as stated above, are called*switching**functions*. The basic switching functions are AND, OR, NOT, NAND, NOR, XOR and XNOR, respectively. Complex logic functions can be obtained by combinations of these basic functions. It may also be noticed that all the other functions can be generated by using NAND or NOR functions alone.

**POSITIVE AND NEGATIVE LOGIC SCHEMES**

As stated above, in all these operations, the associated variables can assume only two values, viz.,

**0**and**1**. We also have stated that**0**s and**1**s can be represented by ON/OFF switches. We know that when an NPN transistor is off, its output will be at +*V*_{CC}and this may be used to represent logic**1;**when it is turned on, the output of the transistor will be 0 volt, and this may be used to represent logic**0**. This logic scheme is known as the*positive-logic scheme*

If we use is a PNP transistor, then the on/off conditions reverse. When it is off, the output will be

‒

*V*_{CC}and this may be treated as logic**0**. When it is in the on state, the output of the transistor is at 0 volt, and this can be used to represent logic**1**. This logic scheme is known as the negative*-logic scheme*. In this textbook, bold numbers**0**and**1**will be used to represent logic**0**and logic**1**, respectively.**The Positive-Logic System**

Figure shows the pulse waveform used to define the positive-logic system (PLS). In PLS, we represent logic

**0**by a voltage that remains at 0-volt level. In other words, in PLS
0 volt ≡

**0**
where we have used a bold

**0**to represent logic 0. Similarly, in PLS, logic**1**is represented by a voltage of +*V*volts. That is,
+

*V*volts ≡**1**
where we have used a bold

**1**to represent logic 1. Here, a positive transition from 0 volt to +*V*volts indicates the logic transition from bit**0**to bit**1**. As stated earlier, NPN transistors can act as switches that perform positive-logic functions.

**The Negative-Logic System**

There are two formats in the negative-logic system

**(NLS). Figure shows the pulse waveform defining Format I of NLS. As shown,****0**is represented by a voltage of +*V*volts. That is,
+

*V*volts ≡**0**
Similarly, in Format I, a voltage level of 0 volt represents

**1.**Thus
0 volt ≡

**1**
In Format II of NLS, we use a negative-voltage swing to represent negative logic. Figure represents this scheme. In this,

**1**is represented by a voltage of 0 volt and**0**is represented by a voltage of −*V*volts. Thus in NLS (Format II),
0 volt ≡

**1**
−

*V*volts ≡**0**_{ }
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