Page 79 - ComputerScience_Class_11

Two more logic gates are obtained when AND and OR gates are inverted. They are named as NAND and NOR gates
respectively. They are also called universal gates because any digital circuit can be implemented using NAND and NOR
gates only.

3.7.4 NAND Gate
The NAND gate is the complement of AND gate. It produces low output (0) only when all the inputs are high (1). In all
other cases, high output (1) is produced. The symbol and truth table of the NAND gate are shown below:
A
A A . B B A.B.C
B C
Symbol of NAND gate Symbol

Truth table of a two-variable NAND gate Truth table of a three-variable NAND gate

A B (A.B)' A B C (A.B.C)'
0 0 1 0 0 0 1
0 1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 0 0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 0

The NAND gate can be demonstrated with the following logic circuit using AND and NOT gate.

A A.B (A.B)'
B

The NAND gate is a universal gate as it can be used to represent the fundamental gates AND or and NOT.

Representing NOT gate using NAND gate only:

When a single Boolean variable passes through the NAND gate it gets complemented as shown in the circuit below:

Output across NAND gate
A A' = (A.A)'
A
= A' [Idempotent Law]

Representing AND gate using NAND gate only:

Two NAND in series are equivalent to AND gate as shown in the logic circuit diagram given below:

Output across NAND gate 1 = (A.B)'
A (A.B)' A.B
1 2 Output of NAND gate 2 = ((A.B)')'
B = A.B [Involution law]

Propositional Logic, Hardware Implementation, Arithmetic Operations 77