Draw the Karnaugh map and find the simplified Boolean equation for the truth table with output 1s for these inputs: ABCD = 0011, ABCD = 0110, ABCD = 1000, ABCD = 1100, and 0s for t... Draw the Karnaugh map and find the simplified Boolean equation for the truth table with output 1s for these inputs: ABCD = 0011, ABCD = 0110, ABCD = 1000, ABCD = 1100, and 0s for the other inputs.
Understand the Problem
The question asks to create a Karnaugh map (K-map) from a given truth table and then derive the simplified Boolean equation from the K-map. The truth table is defined by specifying the input combinations (ABCD) that result in an output of 1. The goal is to use the K-map to identify groupings of these 1s that lead to a simplified Boolean expression.
Answer
$\overline{A}$
Answer for screen readers
$\overline{A}$
Steps to Solve
-
Construct the K-map
A Karnaugh map is a grid that represents the truth table. For four variables (A, B, C, D), we need a $4 \times 4$ K-map. The rows represent AB (00, 01, 11, 10) and the columns represent CD (00, 01, 11, 10).
-
Populate the K-map with 1s based on the given input combinations
The problem states the output is 1 for the following input combinations:
- 0000
- 0001
- 0010
- 0011
- 0100
- 0101
- 0110
- 0111
- 1000
- 1001
- 1010
- 1011
These correspond to the following cells in the K-map (where rows are AB, columns are CD):
- AB = 00, CD = 00 (0000)
- AB = 00, CD = 01 (0001)
- AB = 00, CD = 10 (0010)
- AB = 00, CD = 11 (0011)
- AB = 01, CD = 00 (0100)
- AB = 01, CD = 01 (0101)
- AB = 01, CD = 10 (0110)
- AB = 01, CD = 11 (0111)
- AB = 10, CD = 00 (1000)
- AB = 10, CD = 01 (1001)
- AB = 10, CD = 10 (1010)
- AB = 10, CD = 11 (1011)
Place a '1' in these cells of the K-map grid. The remaining cells will be '0'.
-
Identify the largest possible groupings of 1s
Look for groups of 1, 2, 4, 8, or 16 adjacent 1s, where adjacency includes wrapping around the edges of the K-map. In this case, all cells with AB = 00, 01, and 10 are 1. You can form a group of 12 ones.
-
Derive the simplified Boolean expression from the groupings
The group of 12 ones covers the AB combinations of 00, 01, and 10, which means A's value is changing (from 0 to 1), so A is eliminated. However, B is always 0 or 1 within this group, so B is preserved, hence $\overline{A}$. The group includes all combinations of C and D; therefore C and D are eliminated. Thus, the simplified boolean equation is $\overline{A}$.
$\overline{A}$
More Information
A Karnaugh map is a visual tool used to simplify Boolean algebra expressions. In this instance, the resulting simplification shows that the output is solely dependent on whether the A input is zero (0).
Tips
A common mistake is not identifying the largest possible groupings. For example grouping 2 ones instead of 4 or 8, which leads to a more complex, unsimplified expression. Another mistake is not correctly reading the K-map and misinterpreting which variables are constant within a group.
AI-generated content may contain errors. Please verify critical information
