Boolean Algebra

[shields]

Edited, I fundamentally misunderstood the concept of what I was trying to achieve. Thanks for your help.

 

[Ratchit]

shields,

You made a mistake in reducing the first K-map. There is no b2=w' . It would take 8 minterms to do that. You should use a logic reduction program like Logic Friday - Downloads .

Also change your reductions into OR logic to see if fewer gates are possible.

Ratch

 

[shields]

Yeah that first map is terrible. Im unsure what you mean by change reductions into OR logic? how do i do this? And the program I started to use it, is it only useful for reducing expressions?

*Map should be fixed, thanks for spotting that.

 

[Ratchit]

shields,

Im unsure what you mean by change reductions into OR logic?

A Karnaugh map (K-map) is normally read as the sum of products (SOP). Any logic expression given in SOP terms can be converted into a product of sum (POS) terms. Let's take an example.

Suppose we have A'B'+AC' in SOP format. If we plot it out on a K-map we can see that the minterms are ∑(0,1,4,6). The negative of that expression are the missing minterms, which are ∑(2,3,5,7). Simplifying, we get A'B+AC which is the negative of our original expression. Applying DeMorgans theorem, we remove the negative and get (A+B')(A'+C').


So A'B'+AC' is equivalent to (A+B')(A'+C'). In the first case, all the variables are ANDed to make a term, and in the second case all the variables are ORed to make a term. You should do your reductions both ways to see which reduces to the simplest expression. Note also that all NAND gates, or all NOR gates, or a mix of the two can make any expression you choose. It might not be the least number of gates however. Another trick for an logic expression like AB+BC, which takes two AND gates and one OR gate is to factor it into B(A+C). Then it will only take one AND gate and one OR gate. All these things I pointed out to you can be found in most good logic books.

And the program I started to use it, is it only useful for reducing expressions?

Isn't that what you want? What more do you want it to do?

Ratch

 

[Evan Ariel]

A Boolean algebra is also called a Boolean lattice. The connection to lattices (special partially ordered sets) is suggested by the parallel between set inclusion, A ⊆ B, and ordering, a ≤ b. Consider the lattice of all subsets of {x,y,z}, ordered by set inclusion. This Boolean lattice is a partially ordered set in which, say, {x} ≤ {x,y}. Any two lattice elements, say p = {x,y} and q = {y,z}, have a least upper bound, here {x,y,z}, and a greatest lower bound, here {y}. Suggestively, the least upper bound (or join or supremum) is denoted by the same symbol as logical OR, p∨q; and the greatest lower bound (or meet or infimum) is denoted by same symbol as logical AND, p∧q.
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