Question

Please explain the Cloude Shannon's expansion theory in detail

Answer 1

Shannon's Expansion theory states that a boolean function can be represented by the sum of two sub-functions of the original. Claude Shannon was the creator of Shannon's expansion, and is credited for numerous other achievements in the field of Boolean algebra. Rather than explain it pedantically, here is an example: Take this as function: f= xyz+ xyz + x'y'z + xyz + xyz' Now, notice that we can re-write the function in terms of any two variables- namely, a variable and its compliment. Observe Now, to make the final expression equivalentto our composition of functions, simply apply the distribuive theorem to the function about x: And there we have it! We have just expanded the functionfabout the variable x. Now, you might be thinking to yourself "What if there was no x variable in one or more of the terms? Very good question, and one whose solution is not always intuitive. Remember, that in Boolean algebra, you can AND any literal or term with 1, and still achieve the same truth value. With that in mind, let's look at this function: f-yz+xyz' +x'y'z

If we desire to expand around the variable x, we simply don't have enough information in the first termto accomplish this task. So, what do we do? Give up? No! Remember what was said above: AND the literal with , or, in this case (x' + x). f-yz(x'+x)+ xyz'+x'y'z Expand: f- x'yz + xyz +xyz'+r'y'z This function contains the variable about which we want to expand the expression, so now we should have no problem performing the expansion