Exercise DET-03.02 – Algebraic minimization

Prof. Dr. Thorsten Jungmann (last updated 2026-10-19)

Relates to DET-03 Representation and minimization of logic functions

Task

With the theorems of Boolean algebra, you can minimize an expression step by step, without a map. Minimize each of the following functions as far as possible, and name the theorems that you use in each step. Where a function is given as a minterm list, first write down its canonical disjunctive normal form (DNF).

a) $Y_1=\overline{A}\cdot B+A\cdot B$

b) $Y_2=A\cdot \overline{B}\cdot \overline{C}+A\cdot \overline{B}\cdot C$

c) $Y_3(A,B,C)=\sum m(1,5,7)$

d) $Y_4(A,B,C)=\sum m(0,2,4,6)$

Solution

a) $Y_1=B$
b) $Y_2=A\cdot \overline{B}$
c) $Y_3=\overline{B}\cdot C+A\cdot C=C\cdot \left(A+\overline{B}\right)$, with $A\cdot \overline{B}\cdot C$ used twice (idempotence)
d) $Y_4=\overline{C}$, after uniting twice

▶️ Full worked solution