XNOR gate using NAND

As we know, the logical equation of a 2-input XNOR gate is given as below:

                      Y = A (xnor) 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 = (CD    +    A B)           -----   (i)

     where

                      C = A'     and      D = B'

De-Morgan's law states that
                                m + n = (m'n')'

Taking this into account,

                     Y = ((CD)'(AB)')' = ((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.

Figure 1: 2-input XNOR gate implementation using NAND gates

2x1 mux using NAND gates

As we know, the logical equation of a 2-input mux is given as below:

                      Y = (s' A   +    s B)

Where s is the select of the multiplexer.
De-Morgan's law states that
                                m + n = (m'n')'

Taking this into account, here m = s'A  and  n = sB

                     Y = ((s'A)'(sB)')' = ((s' A)'  (s B)')'

Thus, Y is equal to ((s' nand A) nand (s nand B)). No further optimizations seem possible to this logic. Figure 1 below shows the implementation of 2:1 mux using 2-input NAND gates.

Figure 1: 2:1 Mux using NAND gates