NAND Gate

When output of an AND gate is inverted through a NOT gate, the operation is called NAND operation. The logic gate which performs this NAND operation is called NAND gate.

A NOT gate followed by an AND gate makes a NAND gate. The basis logical construction of the NAND gate is shown below,

The symbol of NAND gate is similar to AND gate but one bubble is drawn at the output point of the AND gate, in the case of NAND gate. The symbol of NAND gate is shown below.

NAND gate means “not AND gate” which means, the output of this gate is just reverse of that of a similar AND gate. We know that the output of the AND gate is only high or 1 when all the inputs are high or 1. In all other cases, the output of AND gate is low or 0. In the NAND, the fact is an opposite, here, the output is only logical 0 when and only when all inputs of the gate are 1s, and in all other cases, the output of NAND gate is high or 1.
Hence, truth table of a NAND gate can be written like,

Just reverse of the truth table of AND gate which is

Like AND gate a NAND gate can also be more than two inputs, like 3, 4, input NAND gate.
A NAND gate is also referred as universal logic gate as all the binary operations can be realised by using only NAND gates.
There are three basic binary operations, AND, OR and NOT. By these three basic operations, one can realise all complex binary operations. Now, we will show how we can achieve all these three binary operations by using only NAND gates.

Realizing NOT Gate Using NAND Gate

When both inputs of a two inputs NAND gate are zero, the output is 1, and both inputs of the NAND gate are 1, the output is 0. Hence a NOT gate can very easily be realised from NAND gates just by applying common inputs to the NAND gate or by short-circuiting all the inputs terminals of a NAND gate.

Where, X is either 1 or 0.

Realizing AND Gate Using NAND Gate

As we told earlier, a NAND gate is a NOT gate followed by an AND gate, so if we can cancel the effect of NOT gate in a NAND gate, it will become an AND gate. Hence, a NOT gate followed by a NAND gate realises an AND gate. In this case, we use the NOT gates realised from NAND gates, and we are showing the logic circuit below,

Realizing OR Gate from NAND Gate

From De Morgan Theorem we know,

The above equation is a logical OR operation.

The above logic equation can be represented by gates as shown above, where inputs first inverted then passed through a third NAND gate.
The truth table of such circuit is,

Now, we have proved that all three basic binary operations can be realized by using only NAND gates. Hence, any other simple or complex binary operation must also be realized by using only NAND gates and hence it is justified to call an NAND gates as universal gates.