Below is a question in my homework:

Implement <span class=\$F(P,Q,R,S,T)=(P+Q)S+(R+T)\bar S\$ using one or more 2x2 AOI." />

Here's my attempt for (a):

Step 1: simplify boolean expression.


\$=(P+Q)S+(R+T)\bar S\$

\$=PS+QS+R\bar S+\bar ST\$

Step 2: expand boolean expression so that it fits into the AOI gate logic:

\$F=PS+QS+R\bar S+\bar ST\$

\$=\overline{\overline{(PS+QS)+(R\bar S+\bar ST)}}\$

\$=\overline{(\overline{PS+QS})(\overline{R\bar S+\bar ST})}\$

I want to know:

  1. Do I continue manipulating the boolean expression, or can I just implement the result from Step 2 with AOI gates?

  2. Is this double-negate manipulating method suitable for implementing boolean functions with AOI gates (or maybe even all gates in general)?

Any guidance is appreciated!


1 Answer 1


The result of Step 2 has obvious two 2X2 AOI. One for the left hand side let us call it X and one for right hand side let's call it Y.

\begin{equation} X = \overline{PS + QS} \; (one AOI22)\\ Y = \overline{R\overline{S} + \overline{S}T} \; (one AOI22) \end{equation}

Also you can implement the inverter using AOI22 as below: \begin{equation} \overline{(S.1 + 0.0)} = \overline{S} \; (one AOI22) \end{equation}

What you need is \begin{equation} \overline{X} + \overline{Y} \end{equation}

which you can perform by one 2x2 AOI gate as:

\begin{equation} \overline{(X.Y + 0.0)} = \overline{X} + \overline{Y} \; (one AOI22) \end{equation}

This makes 4 AOI22 for the result of Step 2.

  • \$\begingroup\$ Thanks for the help!! I forgot that an AOI can be used as an inverter as well, which was what got me stuck. \$\endgroup\$
    – Vero
    Oct 13, 2019 at 14:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.