DEMUX vs decoder; seven-segment "b" segment

DEMUX vs decoder

A demultiplexer sends one data input to one of $2^n$ outputs chosen by $n$ select lines. A decoder converts an $n$-bit input code into one active line out of $2^n$ lines. A decoder with an enable input works as a DEMUX if the enable is used as the data input.

Seven-segment decoder

A BCD-to-seven-segment decoder (e.g. 7447/7448) takes a 4-bit BCD digit $ABCD$ and produces seven outputs $a$ to $g$ that light the segments of an LED display so that the decimal digit 0–9 is shown. Inputs 1010–1111 never occur, so they are don't-cares.

     a
    ---
 f | g | b
    ---
 e |   | c
    ---
     d

Simplest expression for segment "b"

Segment b is ON for digits 0, 1, 2, 3, 4, 7, 8, 9 and OFF for 5 and 6 (standard 7447 style).

$$b = \Sigma m(0,1,2,3,4,7,8,9) + d(10,11,12,13,14,15)$$

K-map (rows AB, columns CD):

Groups: rows 00 and 10 (all $B=0$) give $B'$; column 00 gives $C'D'$; column 11 gives $CD$.

$$b = B' + C'D' + CD = B' + (C \odot D)$$

Circuit

 B ---[NOT]--- B' ---------+
                           +--[OR]--- b
 C --+                     |
     +--[XNOR]--- C⊙D -----+
 D --+

So segment b needs one inverter, one XNOR and one 2-input OR gate (or with AND/OR: $C'D'$ and $CD$ from two AND gates and a 3-input OR).