Question

How will you obtain OR, AND gates from the NAND and NOR gates? Write symbols, Boolean formula, and truth table.

Answer 1

Hint : We will use a NOT gate made from one of the universal gates for the above conversions. In case of obtaining an AND gate from the NAND gate, we will use a NOT gate after the NAND gate since then there will be a double negation on NAND which will give us an AND gate. Similar is the case while obtaining an OR gate from NOR gate where we will use a NOT gate (made from a NOR gate) after another NOR gate.

Complete step by step answer:
The symbol and truth table of the AND gate is

Truth Table

Input A	Input B	Output
0	0	0
0	1	0
1	0	0
1	1	1

The symbol and truth table of the NAND gate is

Truth Table

Input A	Input B	Output
0	0	1
0	1	1
1	0	1
1	1	0

The symbol and truth table of the OR gate is

Truth Table

Input A	Input B	Output
0	0	0
0	1	1
1	0	1
1	1	1

The symbol and truth table of the NOR gate is

Truth Table

Input A	Input B	Output
0	0	1
0	1	0
1	0	0
1	1	0

It is evident from the above truth tables that AND is the negation of NAND and NOR is the negation of OR.
The arrangement of the required gates is shown in the figure. We will obtain the required OR and AND gates from the NOR and NAND gates as illustrated below.

The Boolean formula for the above conversions is as follows, considering the inputs of the gates as $X$ and $Y$:
OR gate from NOR gate: $\overline {\overline {X + Y} } = X + Y$
OR gate from NAND gate: $\overline {\bar X \cdot \bar Y} = X + Y$
AND gate from NOR gate: $\overline {\bar X + \bar Y} = X \cdot Y$
AND gate from NAND gate: $\overline {\overline {X \cdot Y} } = X \cdot Y$

Note:
The above figure is the least number of gates required for the given conversions. If we use more gates, then they will be redundant. Example: we can use two NOT gates (made from NAND or NOR) after any of the above gates but they will be redundant as the two NOT gates will nullify each other.