Introduction
To simplify the Boolean function F(A, B, C, D) = ABC’D’ + ABC’D + ABCD’ + AB’CD’ using a Karnaugh Map (K-map), we will first list the minterms and then proceed with the simplification process.
Minterm Identification
The given function can be represented in terms of minterms:
- ABC’D’ corresponds to m(4)
- ABC’D corresponds to m(5)
- ABCD’ corresponds to m(6)
- AB’CD’ corresponds to m(2)
Thus, the function covers minterms 2, 4, 5, and 6.
Karnaugh Map Setup
We will create a 4-variable K-map:
CD
00 01 11 10
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AB 00 | 0 | 0 | 0 | 0 |
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01 | 1 | 0 | 0 | 1 |
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11 | 1 | 1 | 0 | 1 |
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10 | 0 | 0 | 0 | 0 |
Grouping the Ones
- Group 1: m(4) and m(5) can be combined (vertical pair).
- Group 2: m(6) and m(2) can be combined (horizontal pair).
Simplification
From the K-map:
- Group 1 (m(4), m(5)): common terms give AB'C' (covers D’).
- Group 2 (m(2), m(6)): common terms give ACD' (covers B’).
Thus, the simplified expression is:
F(A, B, C, D) = AB'C' + ACD'
Conclusion
The simplified Boolean function for F(A, B, C, D) is:
F(A, B, C, D) = AB'C' + ACD'
This concise representation reduces complexity and is easier to implement in digital circuits.