Logic and Imagination: Exploring Digital Logic, Boolean Algebra, and Karnaugh Maps

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

2. Digital Logic • Represents Binary outcomes • statement TRUE FALSE • answer YES NO • light OFF On • switch CLOSED OPEN • one bit 1 0

3. 10. A + AB = A 11. A + A’B = A + B 12. (A + B)(A + C) = A + BC Note: A can represent a single variable or a combination of variables 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 9. A’’ = A

4. DeMorgan’s Rules • A + B = (A’B’)’ • AB = (A’+B’)’

5. Gray Code • unsigned decimal gray • 000 0 000 • 001 1 001 • 010 2 011 • 011 3 010 • 100 4 110 • 101 5 111 • 110 6 101 • 111 7 100

6. Karnaugh Maps2 & 3 Variables

7. Karnaugh Maps 4 Variables

8. Karnaugh Map ExampleA’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. 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. 0 0000 1 1 0001 0 2 0010 1 3 0011 1 4 0100 0 5 0101 1 6 0110 1 7 0111 1 8 1000 1 9 1001 1 10 1010 x 15 1111 x Don’t Care Segment 1

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