3:8 decoder building F = Σm(1,2,4,7) — multiplexers and decoders
exam standarddecoder plus OR as a canonical implementation
Answer
A′ · B′ · C + A′ · B · C′ + A · B′ · C′ + A · B · C
Why this example is worth doing
Since each decoder output is one minterm, ORing a chosen subset implements any function of those variables directly from its canonical sum — no minimisation required. The page presents it as the brute-force implementation strategy: more hardware than a minimised gate network, but derived mechanically from the truth table, and cheap when one decoder is shared across several output functions.
Try your own input in the Multiplexers & decoders. Multiplexers from 2:1 to 16:1 and decoders from 2:4 to 4:16, with the selection logic derived.
How the answer is reached
Step by step
A′ · B′ · C + A′ · B · C′ + A · B′ · C′ + A · B · C is already a minimum sum-of-products form (4 terms, 12 literals), proved by Quine–McCluskey — no law can shorten it. It is the only minimum-cost form.
A′ · B′ · C + A′ · B · C′ + A · B′ · C′ + A · B · C is already a minimum sum-of-products form (4 terms, 12 literals), proved by Quine–McCluskey — no law can shorten it. It is the only minimum-cost form.
Quine–McCluskey
| # | Term |
|---|---|
| 1 | A′ · B′ · C |
| 2 | A′ · B · C′ |
| 3 | A · B′ · C′ |
| 4 | A · B · C |
| # | Cover |
|---|---|
| 1 | A′ · B′ · C + A′ · B · C′ + A · B′ · C′ + A · B · C |
4 term(s), 12 literal(s); 4 essential prime implicant(s).
Truth table
| # | A | B | C | F |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 |
| 2 | 0 | 1 | 0 | 1 |
| 3 | 0 | 1 | 1 | 0 |
| 4 | 1 | 0 | 0 | 1 |
| 5 | 1 | 0 | 1 | 0 |
| 6 | 1 | 1 | 0 | 0 |
| 7 | 1 | 1 | 1 | 1 |