Boolean algebra -- minterms for 1:1 inputs and outputs

In fact the true answer is:

We first write the short minterm form of a boolean function for "ones" (where X = 1) of the truth table:

$$ X(A) = \sum m(0) $$

In this case we only have one minterm i.e

$$ m_0 = \overline{A} $$

So we can simply write:

$$ X(A) = m_0\\ \boxed{X(A) = \overline{A}} $$

For the other example we can simmilarly write the short minterm form of the boolean function for "zeroes" (where X = 0):

$$ \overline{X}(A) = \sum m(0,1) $$

In this case we have two minterms i.e

$$ m_0 = \overline{A}\\ m_1 = A $$

So we can simply write:

$$ \overline{X}(A) = m_0 + m_1\\ \overline{X}(A) = \overline{A} + A\\ \overline{X}(A) = 1\\ \boxed{X(A) = 0} $$

This is the correct derivation.

Principles described here are from the book:

"Fundamentals of digital electronics and computer design with VHDL (Sandige)", chapter 1.3.