1.5 Knots Karnaugh Map and Boolean algebra (Sop,pos) part1

In the previous chapters, we have done a lot of exercises. Now the algebra tools in our hands are only the Demorgen theorem and the push bubble theorem. Then in the design of the logic gate, there are many times more than one method of implementation, each with different advantages. So how to design the logic gate circuit of the arbitrary. The answer is to use mathematical tools. Mathematical.. Headache. It doesn't matter, just do it, you can learn it.

Note that the form here is used in the same way as in the binary. Easy to understand and not wordy, later forget, come to this look can be remembered.

1. Symbol definition: As with previous rules ~a represents "non" a,a+b for "A or B" (A or B), a*b for "A and B" (A and B)

2. Priority: And subtraction analogy, Boolean algebra precedence from high to low for not,and,or.

Example Y=A+BC is equivalent to: y=a+ (BC) instead of y= (a+b) C. So parentheses are used to change the priority of operations.

3.SOP (sumofproduct) Form:

First define the minimum item (minterm): Each line of the truth-table must have an expression that makes the input "true" with and combined ... God horse meaning, see example:

(1) (2)

The combination of the minterm here is in the form of and, observing the input and Minterm entries in two examples, regardless of the input, the minterm is "1" (true).

In (1), when the output is Y is 1, minterm only the form of ~ab, so the expression of the first truth table is Y=~ab, the second example of the expression is

Y=~ab+ab. ---magic, no longer have to look for expressions in one line. It's nothing, it's a math trick. In fact, the principle is this, we analyze the truth table

, just find the input combination (and combination) when the output is 1. Because the truth table enumerates all inputs for and, only the values conforming to Y can be selected, and the logical operation is only not,and,or. And in a logical circuit, there can be many input combinations whose output is 1, so as long as these conditions are (plus) or you can get an expression.

and (2) The resulting expression is called the SOP specification form (sum of product) ~ ~ Used to bluff the name of the person. Because the meaning of the formula is to combine the results of two and or (add two product).

Do you think the probability of the operation here? Independent event of God Horse. Yes, it has a close connection! Because the binary number operation itself is a probability operation. For example, there are 8 empty bits, then we have 8 positions to put 0 and 1, then the total probability is

2^8=256, that is, there are 256 ways to put, this is not the range of 8bits (0~255). So the binary operation conforms to the probability calculation. As an example, Y=~A~B~C+A~B~C+A~BC is also the SOP form. (People change their truth-table, and then verify the above.) It feels so long, it's a lot of trouble to draw a line map.

Yes, we're not exploring a shortcut.

4 POS (productofsum) Form:

Same as SOP, but first define maximum (Minterm): Each line of the truth-table must have an expression that makes the input "false" with or combined (principle: the truth table enumerates all the inputs for or, only the values that match Y can be selected), see Example:

。 We use the above SOP's idea to analyze it in step. I will not wordy, do not know the message. The expression for this truth-table is y= (a+b) * (~a+b).

5. Boolean algebra (Boolean Algerbra)

is similar to the general subtraction, only more simple than that. So do not be afraid of OH ~. First I want to introduce the difference between axioms and theorems: axioms cannot be proved, and theorems can be proved by a bunch of axioms.

The second is the definition of double pairs: if 1 and 0,and and or are exchanged at the same time, the result is unchanged. We say that the result has a double-pair nature. For example: If the ~b=1 is b=0 and if ~b=0 b=1 these two descriptions are the same thing, but 1 and 0 are exchanged, that is, the two narratives into a pair of relations.

(If you don't understand it, skip it, then come back and see it after you've learned it.)

(a) Well, now take the definition needed in Boolean algebra:

。 Explain: Axiom is the truth, dual is the double pair that describes the fact from another direction. Name is the name of the axiom.

For example A1 said that if B is not equal to 1, then B equals 0. This is the binary field (see name). The two-pair description is: If B is not equal to 0 B must be 1.a1 ' and A1 very symmetrical bar ~. The rest must be read one after another. It may be confusing to keep talking.

(b) The theorem of only one variable:

。 Here T stands for theorem, which can be proved by the axiom in (a). Note that the identity in name means "own operation (after some sort of operation is the same as the original)"

Mathematics is also called the unit operation. The Null element, as the name implies, is "0 elements", and the concept is 1 like any number of times 0. 0 "and" any input is 0, and 1 or any input is 1. Idempotency is the power of idempotent. That is, an input and self to get is still himself, in and a bit or himself, n times after the b^n=b or himself is the same whether the product is a few times the operation is equal (exponentiation is power). Involution we have already seen, is to seek two times reverse operation. Complements is complementary, an input and its own complementary result must be 0. Because 1 of the complementarity is 0,0 complementary is 1.

Well, here's the first one, and we'll see how we can deal with these tools and more variables.

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