Let us now prove that a positivelogic gate is equivalent to its negativelogic complementary gate. For example, a positivelogic OR gate is equivalent to a negativelogic AND gate, and vice versa. These statements are based on the De Morgan’s laws.
Equivalence of PL OR and NL AND Functions
we had discussed the operation of a twoinput OR gate with the help of Table 2.2. We now modify Table 2.2 to get Table 2.6, which explains the operation of the circuit with the help of positivelogic voltage levels. We now define 0volt level as binary 0 and +V = 5volt level as binary 1 in the positivelogic system. In Col. 1 (i.e., column 1) of Table 2.6, we have entered the input voltage levels of switch A. In Col. 2, we have entered the binary logic levels corresponding to the entries in Col. 1. Thus, in the first row of Col. 1, we have input A = 0 volt. The binary level of 0 volt is bit 0 in positive logic and this is the entry in the first row of Col. 2. This suggests that wherever the input entry is shown as 0 volt, its corresponding binary entry is bit 0.
In the same way as described above, in a positivelogic system, wherever the entry is +5 volt, we have its corresponding binary entry as bit 1. Thus in the third row of Col. 1, we find that input switch A is now applied with a voltage of +5 volts. Then, its corresponding binaryequivalent entry in the third row of Col. 2 is bit 1.
Based on the principles described in the above paragraphs, we complete the entries of Table 2.6. By checking the entries of columns 5 and 6 of this table, we find that the circuit shown, indeed performs OR function in the positivelogic system. Hence, the question mark (?) shown in brackets in the heading of Col. 6 of Table 2.6 may be replaced with a plus (+) sign.
Table 2.6 Positivelogic OR gate
Switch A

Switch B

A (?) B
 
Voltage
level

Binary equivalent

Voltage
level

Binary equivalent

Voltage
level

Binary equivalent

Column 1

Column 2

Column 3

Column 4

Column 5 5

Column 6

0 V
0 V
+5 V
+5 V

0
0
1
1

0 V
+5 V
0 V
+5 V

0
1
0
1

0 V
+5 V
+5 V
+5 V

0
1
1
1

We now prove that the circuit shown, performs AND operation in the negativelogic system. For this, we first modify Table 2.6 by using voltages to represent the negativelogic levels. The modified truth table is shown in Table 2.7.
In Table 2.7, we find that entries in all the columns under the heading voltage level are exactly the same as those found in Table 2.6. However, the entries in the columns of Table 2.7 under the heading binaryequivalent level differ from those given in Table 2.6. For example, in Table 2.7, we put bit 0 where the voltage level is +5 V and bit 1 where the voltage level is 0 V. Now, inspecting Cols. 5 and 6 of Table 2.7, we find that the circuit shown, indeed is performing an AND operation in the negativelogic scheme. Hence, the question mark (?) shown in brackets in the heading of Col. 6 of Table 2.7 may be replaced with a dot (∙) sign.
Table 2.7 Negativelogic AND gate
Switch A

Switch B

A (?) B
 
Voltage
level

Binary equivalent

Voltage
level

Binary equivalent

Voltage
level

Binary equivalent

Column 1

Column 2

Column 3

Column 4

Column 5

Column 6

0 V
0 V
+5 V
+5 V

1
1
0
0

0 V
+5 V
0 V
+5 V

1
0
1
0

0 V
+5 V
+5 V
+5 V

1
0
0
0

Based on the arguments given above, we conclude that positivelogic OR gate is equivalent to negativelogic AND gate and vice versa. Similarly, we can prove that positivelogic NAND gate is equivalent to negative logic NOR gate and vice versa.
Reference:
Table 2.1  AND Function
A

B

Z = AB

0
0
1
1

0
1
0
1

0
0
0
1

Table 2.2  OR Function
A

B

Z = A + B

0
0
1
1

0
1
0
1

0
1
1
1

Table 2.3  NOT Function
A

A′

0
0

0
1

Table 2.4  NAND Function
A

B

Z = (A B)′

0

0

1

0

1

1

1

0

1

1

1

0

A

B

Z = (A + B)′

0

0

1

0

1

0

1

0

0

1

1

0

0 on: "Equivalence of Positive and Negative Logic Systems"