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Week 3

Week 3. Logic will get you from A to B. Imagination will take you everywhere. Albert Einstein. Digital Logic. Represents Binary outcomes statement TRUE FALSE answer YES NO light OFF On switch CLOSED OPEN one bit 1 0. 11. A + A’B = A + B 12. A(B+C) = AB + AC

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Week 3

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  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. 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. 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

  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 • 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 abcd Segment 1

  14. End

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