1. Week 3
Logic will get you from A to B. Imagination will take you everywhere.
Albert Einstein

2. Represents Binary outcomes
Digital Logic
Represents Binary outcomes
statement TRUE FALSE
answer YES NO
light OFF On
switch CLOSED OPEN
one bit

3. Basic Rules of Boolean Algebra
1. A + 0 = A
2. A + 1 = 1
3. A • 0 = 0
4. A • 1 = A
5. A + A = A
6. A + A’ = 1
7. A • A = A
8. A • A’ = 0
A’’ = A
A + AB = A
11. A + A’B = A + B
12. A(B+C) = AB + AC
13. (A+B)(C+D) =
AC+AD+BC+BD
14. (A + B)(A + C) = A + BC
Note: A,B,C can represent a single variable or a combination of variables. Thus, rule 13 can be easily derived from rule 12.

4. DeMorgan’s Rules (A + B)’ = A’B’ (AB)’ = A’ + B’
By taking the inverse of each side they can be re-written as:
A + B = (A’B’)’
AB = (A’+B’)’

5. Gray Code unsigned decimal gray 000 0 000 001 1 001 010 2 011

6. Karnaugh Maps 2 & 3 Variables

7. Karnaugh Maps 4 Variables

8. Karnaugh Map Example A’B’C’ + AB’C’ + A’BC’ + ABC’

9. Karnaugh Map Grouping

10. Karnaugh Map Grouping

11. Karnaugh Map Example Cont. A’B’C’ + AB’C’ + A’BC’ + ABC’
B is not covered, and both B and B’ are included, So we ignore B
C’ is common to the entire grouping, So it is included
A is covered
over the full
Range so we
ignore A
Final Result : X = C’

12. Canonical Form
Canonical means all variables are represented in each term.
X = a’b + ac is a minimum representation
Change to Canonical Form
= a’b(c+c’) + a(b+b’)c
= a’bc + a’bc’ + abc + ab’c
This implies that some variables are redundant

13. Don’t Care abcd x x
BCD to 7 segment display Logic
Each segment is controlled by it’s own logic
To reduce the boolean equation in a Karnaugh Map we plot the don’t care states.
If appropriate we can use these to form larger groupings, thus simplifying the logic.
One equation for each segment.
Segment 1

18. End