1.5 Knots Karnaugh Map and Boolean algebra (Sop,pos) part2

We come back and take a serious look at the axioms (AXIOM) and univariate theorems listed above.

(for its proof, interested people can take a look, not interested can skip.) Here I prove (b) the T4: Because of A1 and its double-pair, we are dealing with binary issues. So according to A2 we do two times A2 operation, it is equivalent to the first operation of the A1, the second time A1 ' operation, the result is the original number itself. So the proof//)

So let's continue the multi-variable (multi-input, 2 or more than 2) processing methods.

(c)

Pay attention to dealing with multivariable, as long as you want to add and subtract polynomial algorithms. (That is, the primary school polynomial calculation)

T6 (same as T6 '): addition (or) Exchange law

T7 (same as T7 '): addition (or) binding law

T8: Distribution Law

The following are the unique/////////////////of binary operations

T8 ': Special, is or assigned to and, whereas general algebra operations do not allocate addition to multiplication, only multiply the allocation rate. Can look like this:

T9: Cover Law

T10: Combination Law

T11: Consensus law

T12: That's Playa de Mogan law.

Want to prove these theorems? Yes, that's what we're going to do. Check each one out. Will not prove. Binary is only 1 and 0, the column truth table is the poor lifting method proved. Don't be lazy. A capable person can use some of the theorems in (b) and (c) to push to the theorem you want.

For example: In order to prove T9 ': With T8 ': B + (B*c) = (b+b) * (b+c) = b* (b+c) = B, because B or C, the result can take B and C, if take B to get the result. //

So Morgan's law is almost omnipotent. You can export all of the above theorems (and, of course, combine them with others).

Here I give a proof of the consensus law:

。 It's natural to see the watch.

Someone said, so much, can't remember how to do. Focus on this: there is only one way, but also can make you a master of the beginning: is to take the trouble to use (how to use the following), with the following about the Carnot diagram. If you can do Karnaugh map and the above theorem flexible use, you have been a step away from the master.

(Don't rely on VERILOG,HDL's automatic tools, which is why most people go to yards.) Too automatic, their brains lazy, and later when we design VLSI, of course, to use those tools, but the integration of millions of CMOS, the machine is not so smart, because it does not realize the design of art and intuition. Trust me, Intel and AMD's senior engineers are the ones who can quickly find an optimization strategy in a thousands of CMOS portfolio because they see more and form modules in their brains. So when you have the ability to do this, you can be a master level. ）

So here's the application--the above-mentioned theories, the goal is only one, with less CMOS do more things.

For example: simplification Y=~A~B~C+A~B~C+A~BC

The first method of simplification: (justification lists which theorem is applied)

The second method of simplification: (justification lists which theorem is applied)

Note Here the application of b=b+b+b+ ... Theorem is the double-pair theorem of T3 equal power theorem.

Here, everyone has to write the task is 1: self-proof (c) The correctness of the theorem, not lazy oh

2. Implement the last example with a logical diagram (two simplification methods are needed)

This article is from the "Basic to Master" blog, please be sure to keep this source http://physic.blog.51cto.com/1656469/1306007