Making a logic circuit with only NAND GATES?

Question

I am trying to create a logic circuit using only NAND Gates for this expression: (NOT Q AND P) OR R

This question has really gotten me stuck! Can somebody please help?

Answer 1

The best way for a beginner is to think about each term separately and how you would create that with a NAND gate.

Let's start with the basic NAND gate truth table:

A | B | Q
---------
0 | 0 | 1
1 | 0 | 1
0 | 1 | 1
1 | 1 | 0

Now let's look at each term. We have a NOT, and AND, and an OR in there. So, how can we make those with NAND gates? Start with the NOT.

How does a NAND look like a NOT? Simple - when both inputs are the same. If you tie A and B together so they always see the same signal, then you have a NOT gate. 0 nand 0 = 1, 1 nand 1 = 0. So the NOT gate can just be:

^{simulate this circuit – Schematic created using CircuitLab}

Next the AND. What's a NAND? It's an inverted AND. And AND with a NOT after it. So we just want to get rid of the NOT, and you can do that by adding another NOT:

^{simulate this circuit}

Then comes the OR. Look closely at the NAND table and compare it to the OR table:

A | B | Q
---------
0 | 0 | 0
1 | 0 | 1
0 | 1 | 1
1 | 1 | 1

Do you see a similarity? If you were to invert the A and B values the table would instead look like this:

A | B | Q
---------
1 | 1 | 1
0 | 1 | 1
1 | 0 | 1
0 | 0 | 0

And that's the same sequence for Q as the NAND gate. So an OR gate is just a NAND gate with the inputs inverted. And we know how to invert already. So the OR looks like:

^{simulate this circuit}

So now you know what the gates look like you can put the whole thing together:

^{simulate this circuit}

However there's too many gates there. It can be simplified. The bit I have marked with a box - a NOT followed by a NOT - that's completely pointless (from a logic point of view) and is just wasteful. It serves no purpose. So you can get rid of it. Simplify to:

^{simulate this circuit}

Answer 2

\$(\overline Q P) + R\$

\$=\overline{(\overline {\overline Q P}) \overline R}\$

= (NOT Q NAND P) NAND NOT R