I have the SOP (Sum of Products) equation: F = ¬A¬BC + A¬BC + A¬B¬C + ABC, which is the sum output of a full adder.
Could someone please help me and show me how by using Boolean algebra i can change this SOP expression to use only XOR gates.
In table below CIN is C the from formula of F above and Sum is result of F.
$$
\begin{array}{c|c|c|c|c}
\text{A} & \text{B} & C_{IN} & C_{OUT} & \text{Sum} \\
\hline
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 0 & 1 \\
0 & 1 & 1 & 1 & 0 \\
1 & 0 & 0 & 0 & 1 \\
1 & 0 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
$$
Best Answer
One can not implement every logical function using only XOR gates.
Since XOR is a logical operator which obeys associativity, the function implemented using only XOR gates can always be written as $$f = a_1\oplus a_2 \oplus a_3 \oplus \cdots \oplus a_n $$ where \$a_1, a_2 \ldots\$ are the inputs.
So the only possible functions that can be implemented using XOR gates are:
1. Odd number of 1's
$$\tag1 f= \begin{array}{|cl} 1 & \mathrm{if }\ N = odd \\0 & else \end{array}$$
Where N is the number of 1's in the input. \$f\$ can be implemented as $$f = a_1\oplus a_2 \oplus a_3 \oplus \cdots \oplus a_n $$
2. Even number of 1's
$$\tag2 f= \begin{array}{|cl} 1 & \mathrm{if }\ N = even \\0 & else \end{array}$$
can be implemented as $$f = a_1\oplus a_2 \oplus a_3 \oplus \cdots \oplus a_n \oplus 1 $$
3. Inverter
$$\tag3 f = \overline{a}$$ can be implemented as $$f = a\oplus 1 $$
In your case,
SUMcan be implemented using XOR gates only (as given in (1)) but notCOUT.