Finding minterms from a boolean expression

Question

I have a question regarding the process of finding minterms.

Problem: Find the minterms of the following expression by first plotting each expression on a K-map:

a) F(X,Y,Z) = XY + XZ + X’YZ
b) F(W,X,Y,Z) = XZ + W’XY’ + WXY + W’YZ + WY’Z 
c) F(A,B,C,D) = B’D' + ABD + A’BC

Is there any way that I can find the minterms of function without making a truth table? Like can you just look at the boolean expression and plot the 1's for the minterm on a k-map?

Answer 1

Take $F(A,B,C,D) = BD + \dots$

In the K-map, addition is the union of each terms. The first term being the intersection of $B$ with $D$ (i.e. logically both B and D are true).

The $B$ is represented by two rows, $\bar{A}B$ and $A B$ since $(\bar{A}+A) B = B$. Similarly, $D$ is represented by two rows, as shown.

After we fill the k-map for all terms, we can visually read the minterms from their locations in the k-map. If $F(A,B,C,D) = BD$ as below, then $F = \bar{A}B\bar{C}D + \bar{A}BCD + AB\bar{C}D + ABCD$, (i.e. true for inputs 0101, 0111, 1101, 1111).

Answer 2

You can use boolean algebra simplification laws for reducing variables, but the K-map will give you exactly the minterm, for what I know there is no other way to solve it.