# Implement boolean expression with AOI

Below is a question in my homework:

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

Here's my attempt for (a):

Step 1: simplify boolean expression.

$$\F(P,Q,R,S,T)\$$

$$\=(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!

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.

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

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

What you need is $$$$\overline{X} + \overline{Y}$$$$

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

$$$$\overline{(X.Y + 0.0)} = \overline{X} + \overline{Y} \; (one AOI22)$$$$

This makes 4 AOI22 for the result of Step 2.

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