Page 90 - Computer Science V2.0 Class 11

1. Show the truth table for X-NOR gate.
2. Draw the logic diagram for equivalence gate.

3.4.3 Fundamental Gates

NOR and NAND are called fundamental gates because if we use one of NOR and NAND gates, all other gates may be
implemented using that gate only. We show below how to implement the AND, OR, NOT gates using the NAND gate
only:
AND gate: A ● B = ((A ●B)')' = ((A ● B)' ● (A ● B)')'
= ((A NAND B) NAND (A NAND B)

OR gate: A + B = ((A + B)')' = (A' ● B')'
= ((A NAND A) NAND (B NAND B))

NOT gate: A' = A NAND A
Similarly, each of the three basic gates AND, OR, NOT can be implemented using NOR gate only. Once basic gates are
implemented, all other gates can be implemented using them.

1. Give an expression for AND gate using only NOR gate.
2. Give an expression for OR gate using only NOR gate.
3. Give an expression for NOT gate using only NOR gate.

3.4.4 Boolean Functions Revisited

In the beginning of section 3.4.1, we saw how to compute Boolean functions using a truth table. In this section, given
a truth table, we will derive the corresponding Boolean function. For example, consider Table 3.11, which shows the
truth table for a Boolean function of three variables x, y, z.

x y z f
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
Table 3.11: Truth table for Boolean function of three variables x, y, z
In order to arrive at a boolean function, we first determine the rows for which function takes the value one. There are
three such rows, viz. third, sixth, and seventh. From third row, we find that F should be equal to 1 if x = 0, y = 1,
z = 0. Now we find a boolean expression which yields 1 if and only if x = 0, y = 1, z = 0. Equivalently we
should find an expression which is equal to 1 if and only if x' = 1, y = 1, z' = 1. Clearly, the product x'yz'
meets this requirement. Note that the expression x'yz' will be equal to 1 for x = 0, y = 1, z = 0, and for
no other combination of values of x, y, z. Similarly we find that for sixth row the expression xy'z will be equal to
1 if and only if x = 1, y = 0, z = 1, and for the seventh row the expression xyz' will be equal to 1 if and only
if x = 1, y = 1, z = 0. Next, we note that the boolean function F has to be such as would evaluate to 1 for
any of the combinations (x = 0, y = 1, z = 0), (x = 1, y = 0, z = 1), (x = 1, y = 1, z = 0) and

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