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

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

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.

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’)’

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

Karnaugh Maps 2 & 3 Variables

Karnaugh Maps 4 Variables

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

Karnaugh Map Grouping

Karnaugh Map Grouping

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’

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

Don’t Care abcd 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 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

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