Math Problem Statement
Solution
The Boolean expression provided is:
This is a sum of minterms representation, where the given minterms correspond to specific binary combinations of the variables , , , and .
Step 1: Generate the Karnaugh Map (K-Map)
We can simplify this Boolean function by placing these minterms into a 4-variable K-map and applying grouping to simplify the expression.
The binary representations of the minterms are:
- m(1) = 0001
- m(3) = 0011
- m(7) = 0111
- m(8) = 1000
- m(10) = 1010
- m(12) = 1100
- m(13) = 1101
- m(15) = 1111
We will now populate the K-map.
AB\CD | 00 | 01 | 11 | 10 |
---|---|---|---|---|
00 | 0 | 1 | 1 | 0 |
01 | 0 | 0 | 1 | 1 |
11 | 1 | 1 | 1 | 0 |
10 | 1 | 0 | 1 | 0 |
Step 2: Simplification Using the K-Map
From the K-map, we can group adjacent 1's:
- Group 1 (4 cells): m(12), m(13), m(15), m(7) →
- Group 2 (2 cells): m(8), m(12) →
- Group 3 (2 cells): m(3), m(7)
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Math Problem Analysis
Mathematical Concepts
Boolean Algebra
Karnaugh Maps
Logic Circuit Design
Formulas
Sum of Minterms
Karnaugh Map Simplification
Theorems
Simplification of Boolean Functions
Suitable Grade Level
University/College Level
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