Lab 8 Problems: Simplifying Boolean Formulas

---SIMPLIFICATION VIA BOOLEAN ALGEBRA---

1.) Use boolean algebra to simplify the equation.
SHOW YOUR WORK, one step per line.  You should use only variable names,
negation (with `!`), AND (by putting terms next to each other),
OR (with `+`), and parentheses.  In addition, each line should start
with an equal sign (`=`). The first step has been done for you to show how.
As a hint, problem 1c requires De Morgan's Law.

Example) f(Z,Y,X) = !ZY!X + ZY!X + !YX
= (!Z + Z)Y!X + !YX
= Y!X + !YX

1a) f(A,B,C) = !A!BC + A!B!C + !ABC + !AB!C + A!BC



---SIMPLIFICATION VIA KARNAUGH MAPS (K-MAPS)---

Each of these problems starts with a truth table which you may need to
complete.  From there, you need to write an unoptimized sum-of-products
formula, where each product corresponds to a single 1 in the output of
the table, and all the products are summed together.  You will then use
a K-Map to simplify the formula on paper, and then write out the simplified
formula in this file.

2.) Given this truth table, find the unoptimized formula this corresponds
to, along with the version which was optimized with a K-Map.

  A B C | O
  _________
  0 0 0 | 0
  0 0 1 | 1
  0 1 0 | 0
  0 1 1 | 1
  1 0 0 | 1
  1 0 1 | 0
  1 1 0 | 1
  1 1 1 | 0


Unoptimized Sum of Products equation
2a) O =

Draw the K-map and simplify Optimized Sum of Products equation
2b) O =

3.)  Given this truth table, find the unoptimized formula this corresponds
to, along with the version which was optimized with a K-Map.

  A B C D | O
  ___________
  0 0 0 0 | 1
  0 0 0 1 | 1
  0 0 1 0 | 1
  0 0 1 1 | 1
  0 1 0 0 | 1
  0 1 0 1 | 0
  0 1 1 0 | 0
  0 1 1 1 | 0
  1 0 0 0 | 1
  1 0 0 1 | 0
  1 0 1 0 | 0
  1 0 1 1 | 0
  1 1 0 0 | 1
  1 1 0 1 | 0
  1 1 1 0 | 0
  1 1 1 1 | 0

Unoptimized Sum of Products equation
3a) O =

Draw the K-map and simplify Optimized Sum of Products equation
3b) O =

4.)  Given this truth table, find the unoptimized formula this corresponds
to, along with the version which was optimized with a K-Map.

  A B C D | O
  ___________
  0 0 0 0 | 0
  0 0 0 1 | 0
  0 0 1 0 | 0
  0 0 1 1 | 0
  0 1 0 0 | 1
  0 1 0 1 | 1
  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 | 1
  1 1 0 1 | 1
  1 1 1 0 | 0
  1 1 1 1 | 1

Unoptimized Sum of Products equation
4a) O =

Draw the K-map and simplify Optimized Sum of Products equation
4b) O =



---DON'T CARES IN KARNAUGH MAPS (K-MAPS)---

5.) Given this truth table, find the unoptimized formula this corresponds
to, along with the version which was optimized with a K-Map.

  A B C D | O
  ___________
  0 0 0 0 | 1
  0 0 0 1 | X
  0 0 1 0 | 0
  0 0 1 1 | 1
  0 1 0 0 | X
  0 1 0 1 | 0
  0 1 1 0 | X
  0 1 1 1 | 0
  1 0 0 0 | X
  1 0 0 1 | 0
  1 0 1 0 | 1
  1 0 1 1 | 0
  1 1 0 0 | 1
  1 1 0 1 | 0
  1 1 1 0 | X
  1 1 1 1 | 0

Unoptimized Sum of Products equation
5a) O =

Draw the K-map and simplify Optimized Sum of Products equation
5b) O =