Boolean expression and truth table

Y = \overline{S} \cdot \overline{A + B} + S \cdot (A \cdot B) = \{S = C \oplus D\} = \overline{C \oplus D} \cdot \overline{A + B} + (C \oplus D) \cdot (A \cdot B)

K-Map and simplified expressions

Circuit can be built from SOP:

A	B	C	D	Y
0	0	0	0	1
0	0	0	1	0
0	0	1	0	0
0	0	1	1	1
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	0
1	0	0	0	0
1	0	0	1	0
1	0	1	0	0
1	0	1	1	0
1	1	0	0	0
1	1	0	1	1
1	1	1	0	1
1	1	1	1	0

Interestingly, Logisim does not generate the simplest Boolean expression from the K-Map for POS! One of the two row MAX-terms can be removed if the other groups are adjusted, so for instance

Y = (A + C + \overline{D})(A + \overline{C} + D)(\overline{B} + C + D)(\overline{B} + \overline{C} + \overline{D})(\overline{A} + B)

is simpler and requires one less two-input OR gate.