Showing posts with label XOR gate with NAND gates. Show all posts
Showing posts with label XOR gate with NAND gates. Show all posts

How to build an XOR gate using NAND gates

We can build a 2-input XOR gate using 5 NAND gates. Sound interesting, isn't it? Let us see how.

As we know, the logical equation of a 2-input XOR gate is given as below:
                      Y = A (xor) B = (A' B    +    A B')
Let us take an approach where we consider A and A' as different variables for now (optimizations related to this, if any, will consider later). Thus, the logic equation, now, becomes:
                       Y = (CB    +    A D)           -----   (i)
     where
                      C = A'     and      D = B'
De-Morgan's law states that

                                m + n = (m'n')'

Taking this into account,
                     Y = ((CB)'(AD)')' = ((A' B)'  (A B')')'
Thus, Y is equal to ((A' nand B) nand (A nand B')). No further optimizations to the logic seem possible to this logic. Figure 1 below shows the implementation of XOR gate using 2-input NAND gates.
A 2-input XOR gate can be implemented as shown in figure.
Figure 1: 2-input XOR gate implementation using 2-input NAND gate
Thus, we have seen an XOR gate can be implemented by putting NAND gates in cascade. Can you think of a better way of implementing XOR gate using NAND gates?


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