I am self-studying the Nand2Tetris course. I am trying to simplify the Or logic gate as much as possible to simplify my HDL-specified circuit.
Using the Sum of Products, I write the following for the Or truth table:
(~A.B) + (A.~B) + (A.B)
= (~A.B) + (A).(B + ~B)
= (~A.B) + (A).(0)
= (~A.B) + 0
= ~A.B
However, the truth table of ~A.B isn't the same as the Or truth table, so I'm doing something wrong. I'm just not sure what it is. Please help : )
Recall that the statement "$B$ is true or $B$ is false" is a tautology, so you should have gotten: \begin{align*} \overline AB + A\overline B + AB &= \overline AB + A(\overline B + B) \\ &= \overline AB + A(1) \\ &= \overline AB + A \\ \end{align*}