What is the equivalent gate for the following digital circuit?
Show Hint
A NAND gate with both inputs connected together behaves as a \textbf{NOT gate}.
Use De Morgan's theorem to simplify combinations of NAND/NOR gates quickly.
Concept:
Use properties of universal gates and De Morgan's theorem:
\[
(A+B)' = A'B'
\]
Also note that a NAND gate with both inputs same acts as a NOT gate.
Step 1: Analyze the first stage.}
Each input passes through a NAND gate whose inputs are tied together.
Thus,
\[
X \rightarrow (X \cdot X)' = X'
\]
\[
Y \rightarrow (Y \cdot Y)' = Y'
\]
So outputs of first stage are:
\[
X',\; Y'
\]
Step 2: Second stage NAND operation.}
These inputs go to a NAND gate:
\[
(X' \cdot Y')'
\]
Using De Morgan's theorem:
\[
(X'Y')' = X + Y
\]
Step 3: Final gate.}
The last gate is again a NAND with both inputs tied together, which acts as a NOT gate:
\[
(X+Y)'' = X+Y
\]
Thus the final output becomes:
\[
X + Y
\]
which is the function of an OR gate.
