F = A’B’C’ + A’BC’ + ABC’ truth table
kmap
The Correct Answer and Explanation is :
To find the simplified Boolean expression for the given equation ( F = A’B’C’ + A’BC’ + ABC’ ), we can follow these steps:
1. Construct the Truth Table
We first construct a truth table to evaluate the expression ( F = A’B’C’ + A’BC’ + ABC’ ). The variables involved are ( A ), ( B ), and ( C ), and each combination of these variables gives a different output for the function ( F ).
| A | B | C | A’B’C’ | A’BC’ | ABC’ | F (A’B’C’ + A’BC’ + ABC’) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
2. K-map (Karnaugh Map) Method
Next, we use a Karnaugh map to simplify the expression. A Karnaugh map is a graphical method of simplifying Boolean functions. Here’s the K-map for 3 variables:
| AB \ C | 0 | 1 |
|---|---|---|
| 00 | 1 | 0 |
| 01 | 1 | 0 |
| 11 | 1 | 0 |
| 10 | 0 | 0 |
3. Simplification
From the K-map, we can group the 1s into the smallest number of groups. We observe the following:
- There is a group of two 1s in the first column (where ( C = 0 ) and ( A’B’ ) and ( A’B ) are the conditions).
- There is a group of two 1s in the second row.
The simplified Boolean expression from this grouping is ( F = A’C’ + BC’ ).
4. Final Answer
The simplified Boolean expression for the given function is ( F = A’C’ + BC’ ).
Explanation
- Truth Table: The truth table evaluates the original equation for all combinations of the variables ( A ), ( B ), and ( C ). This ensures that we account for all possible scenarios.
- K-map: The K-map technique allows us to visually identify which terms can be grouped together and eliminated to simplify the Boolean expression. By identifying the patterns in the K-map, we reduce the original function to a more manageable form.
- Simplification: The simplification process reduces the expression from three terms to two terms. By using a Karnaugh map, we can make the expression more efficient and easier to implement in logic circuits.