Question

I am unsure how to use the Distributive property on the following function:

F = B'D + A'D + BD

I understand that F = xy + x’z would become (xy + x’)(xy + z) but I’m not sure how to do this with three terms with two variables.

Also another small question:
I was wondering how to know what number a minterm is without having to consult (or memorise) the table of minterms.

For example how can I tell that xy’z’ is m4?

Answer 1

When you’re trying to use the distributive property there, what you’re doing is converting minterms to maxterms. This is actually very related to your second question.

To tell that xy’z’ is m4, think of function as binary where false is 0 and true is 1. xy’z’ then becomes 100, binary for the decimal 4. That’s really what a k-map/minterm table is doing for you to give a number.

Now an important extension of this: the number of possible combinations is 2^number of different variables. If you have 3 variables, there are 2^3 or 8 different combinations. That means you have min/maxterm possible numbers from 0-7. Here’s the cool part: anything that isn’t a minterm is a maxterm, and vice versa.

So, if you have variables x and y, and you have the expression xy’, you can see that as 10, or m2. Because the numbers go from 0-3 with 2 variables, m2 implies M0, M1, and M3. Therefore, xy’=(x+y)(x+y’)(x’+y’).

In other words, the easiest way to do the distributive property in either direction is to note what minterm or maxterm you’re dealing with, and just switch it to the other.

For more info/different wording.