How do I find the Boolean equation from a K-map with one '1' and the rest 'don't cares'?

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to find the Boolean equation from the K-map:

Identify the values in the K-map.
The K-map shows three variables: $Q_A$ , $Q_B$ , and $Q_C$ .
The columns represent $Q_A$ (1 and 0).
The rows represent $Q_B Q_C$ (00, 01, 11, 10).
The cell at $Q_A=1, Q_B Q_C=00$ contains a '1'.
All other cells contain 'X', which typically denotes a 'don't care' condition in K-maps. We can use 'don't cares' to form larger groups and simplify the expression.
Group the '1's and 'don't cares' (X's) to form the largest possible groups of powers of 2.
The '1' at $(Q_A=1, Q_B=0, Q_C=0)$ must be covered.
We can form a group of 4 cells by including this '1' and the three 'X's in the same column where $Q_A=1$ .
The cells in this group are:
- $Q_A=1, Q_B=0, Q_C=0$ (value 1)
- $Q_A=1, Q_B=0, Q_C=1$ (value X)
- $Q_A=1, Q_B=1, Q_C=1$ (value X)
- $Q_A=1, Q_B=1, Q_C=0$ (value X)
Write the Boolean expression for the formed group.
For this group of 4 cells:
- $Q_A$ is constant at 1.
- $Q_B$ changes from 0 to 1 within the group, so it is eliminated.
- $Q_C$ changes from 0 to 1 within the group, so it is eliminated. Therefore, the simplified term for this group is $Q_A$ .
Combine the terms.
Since this single group covers the only '1' in the K-map, the simplified Boolean equation is just this term.

The equation is:
F = \boxed{\text{Q_A}}

Answer