# Can these logic circuits be minimised any more?

I have four Boolean functions for four outputs, one for each. I've found the equivalent logic circuits for each in what I believe to be the simplest form but that is what I'm asking. Can these circuits be simplified more in any way so to minimise the chip (logic gate) count?

Not sure how to embed pictures but here are the Booleans: $$a) AD'+AB'C'$$ $$b) ABD'$$ $$c) B'CD'+ACD'$$ $$d)B'C'D$$ I realise I can rewrite a) and c) but do I want to? My aim is to get a circuit with as little logic gates as possible. I was thinking I could use a NOR gate followed by an OR gate for a) to reduce the number of NOT gates I'd need.

• Which gates do you have available? One answer below makes use of three-input AND gates -- is that cheaper for you than two two-input ANDs? If you're counting chips rather than gates, is there a bonus for keeping the number of different kinds of gates low? – Henning Makholm Mar 27 at 23:25
• Well, when I say chips I mean gates, specifically e.g. 7402, 7404 etc. I have the seven basic ones at my disposal: AND, NOT, OR, NOR, NAND, XOR and XNOR. I don't need to factor cost into it either. This is purely for simulation. This question will help me with an overarching problem I haven't described here where bonus marks are available for using a lesser amount of gates. – AustereTiger Mar 27 at 23:29
• A 7402 has four NOR gates in one package. If you have a circuit that uses one NOR gate and another that uses two, would that be a reason to prefer one over the other? You'd need one 7402 in either case. And I don't know what you mean by "don't need to factor cost into it". You can't even start talking about optimizing anything without having a cost function that you want to minimize. – Henning Makholm Mar 27 at 23:37
• Well, I'm not doing the circuit physically. I'm using a simulation package called Multisim which is just used to simulate what a circuit will behave like. Cost doesn't come into it. This isn't real life, per se. Maybe I need to provide some extra context. – AustereTiger Mar 27 at 23:41
• if you have no concept of cost, there is no reason to prefer one circuit to another. Just use the first one that comes to mind. (Note that a cost function does not need to give a value in money -- but unless you have something to define which solutions you would prefer over others, you have not in fact asked a meaningful question). – Henning Makholm Mar 27 at 23:43