Hey there

I'm trying to complete a question which asks me to use the truth table of a made-up universal gate, to build AND, OR and NOT circuits using only that universal gate.

I'm unsure as to how I should go about doing it.

What I have done so far is based on the truth table (2-inputs), I have worked out the boolean equation.

I've been playing around with the universal gate, trying to figure out how I am to get an AND gate but am getting increasingly confused. Are there any steps I should follow to figure out where I am to place the universal gate? I ahve the equation

Any tips? tricks?

 

There are a couple ways to proceed. You know that you can use NAND and NOR as universal gates, correct? So if you can figure out how to make either a NAND or a NOR, you are done. If you can figure out how to make an inverter, then all that is left is to figure out how to make either an AND or and OR. You don't have to make both.

What is the truth table for this made-up universal gate? Given that there are only sixteen possible 2-input Boolean gates and that eight of them (the symmetric ones) are the familiar ones that we use all the time. Of the remaining eight, four implement inverters and buffers one one input or the other and we know these are not universal gates, leaving only four possibilities for your presumed universal gate. Of those, there are two pairs that are mirror images of each other (i.e., the same except that the role of the two inputs are swapped) and the two pairs are logical inverses of each other. Since we are free to swap inputs, that means that we only have to possibilities that we need to consider. Let's call them Bob and Sue.

A|B|BOB|SUE
0|0|0|1
0|1|1|0
1|0|0|1
1|1|0|1

I'm willing to bet that your gate is SUE (or SUE with A and B swapped).

Do you see how you can make in inverter out of a SUE gate?

Now, with an inverter, you can invert any combination of the inputs and outputs, so as long as the basic functionality of the gate can be described using a single AND or a single OR term (with whatever inversions it happens to have on the inputs and outputs), you can use your inverter to undo them and end up with a straight AND or OR with uninverted inputs and outputs. As long as the truth table only has a single 0 or a single 1 in it, you are guaranteed of being able to write it using a single term, so we know we can do this for a SUE gate.

That, plus the inverter, is all you need to be universal. You might even be able to end up with a NAND or NOR even easier, in which case you are done that much quicker.

Since one pair

 

this is what you are looking for i think

 

this is what you are looking for i think

How does a transistor implementation of a NAND and NOR help the OP implement the basic Boolean operations (AND, OR, NO) starting from the truth table of a 2-input Boolean function that is neither NAND or NOR?

 

truth table is

X Y Z
0 0 0
0 1 1
1 0 0
1 1 0

So we're looking at Bob.

 

Do you see how you can make an inverter out of a Bob gate?