The 7-segment decoder example works through segment "a" completely, as a live interactive walkthrough. This is the full companion: all seven segments, each independently minimized and verified, since a real decoder needs every one of them designed, not just one.
The segment map
Standard 7-segment digit patterns, with B3B2B1B0 as the BCD input (labeled A, B, C, D below) and codes 10-15 as don't-cares throughout, since they're invalid BCD and never occur:
| Digit | a | b | c | d | e | f | g |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 2 | 1 | 1 | 0 | 1 | 1 | 0 | 1 |
| 3 | 1 | 1 | 1 | 1 | 0 | 0 | 1 |
| 4 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 5 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 6 | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
| 7 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 9 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
All seven, minimized and verified
What stands out across all seven
Segment e is the simplest — just two terms — while segment d needs five. That variation is expected: each segment is independently minimized from its own truth table, and there's no reason different segments should compress to similar sizes just because they're part of the same display. Segment c reduces to only three single/double-literal terms (C′ + D + B) — worth noticing that it doesn't depend on A at all, meaning segment c's state is fully determined by the lower three BCD bits regardless of whether the digit is 0-7 or 8-9.
Building the complete circuit
Each of the seven expressions above becomes its own independent 2-level AND-OR circuit, all seven sharing the same four BCD input lines (A, B, C, D) but with entirely separate gates — there's no sharing of intermediate results between segments in a basic implementation, though larger-scale designs sometimes factor out common sub-expressions across segments as a further optimization beyond single-output K-map minimization.
See segment a solved interactively in the live example, or verify any of the other six directly on the 4-variable solver.