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D e MORGAN’s LAWS

D e MORGAN’s LAWS. De Morgan’s Laws provide an easy way to find the inverse of a boolean expression: (X + Y)’ = X’ Y’ (X Y)’ = X’ + Y’ An easy way to remember this is that each TERM is complemented, and that OR’s become AND’s; AND’s become OR’s.

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D e MORGAN’s LAWS

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  1. DeMORGAN’s LAWS De Morgan’s Laws provide an easy way to find the inverse of a boolean expression: (X + Y)’ = X’ Y’ (X Y)’ = X’ + Y’ An easy way to remember this is that each TERM is complemented, and that OR’s become AND’s; AND’s become OR’s. (Easy to prove this via a truth table, see textbook p. 47.) The complement of the product is the sum of complements The complement of the sum is the product of complements. -- Can easily generalize to n variables using above rules

  2. APPLYING DeMORGAN’S LAW Apply De Morgan’s Law to a more complex expression: (AB + C’D)’ = (AB)’ (C’D)’ = (A’+B’)(C + D’) Note that De Morgan’s law was applied twice. Another example:[(A’ + B)C’]’ = (A’ + B)’ + (C’)’ = (A’)’ (B)’ + C = AB’ + C

  3. NAND, NOR GATES Why do we care about De Morgan’s Law? There are two other gate types that produce the complement of a boolean function! A B Y0 0 10 1 11 0 11 1 0 A B Y0 0 10 1 01 0 01 1 0 (AB)’ (A+B)’ A A B B NAND NOR

  4. NAND, NOR (cont.) NAND (NOT AND) - can be thought of as an AND gate followed by an inverter. A (AB)’ A AB (AB)’ B B NAND NOR (NOT OR) - can be thought as an OR gate followed by an inverter. A+B A (A+B)’ A (A+B)’ B B NOR

  5. Actually…. In the real world, an AND gate is made from an NAND gate followed by an inverter. An OR gate is made from a NOR gate followed by an inverter. A A (AB)’ AB AB B B NAND AND (A+B)’ A A+B A+B A B B NOR OR

  6. What is this logic function in SOP form? A (AB)’ B F = ((AB)’ (CD)’)’ C D (CD)’ Lets use De Morgan’s Law F = ((AB)’ (CD)’)’ = ((AB)’)’ + ((CD)’)’ = AB + CD An interesting result…………… SOP Form

  7. NAND-NAND form = AND-OR form A (AB)’ B F = ((AB)’ (CD)’)’ C D (CD)’ Same logic function A (AB) B F = AB + CD C D (CD)

  8. nand nor and or not DeMORGAN’S THEOREM • Symbolic DeMorgan’s duals exist for all gate primitives

  9. DeMORGAN’S THEOREM (cont’d) Can save you time in evaluating/designing combinatorial logic – Align bubbles and non-bubbles whenever possible

  10. SHORTCUTS for MULTIPLYING A short cut theorem for Distribution (Multiplication) (X + Y) (X’ + Z) = XZ + X’Y Only works when you have a variable (X) and its complement (X’). To PROVE this, lets do the distribution the long way. (X + Y)(X’ + Z) = X X’ + XZ + X’Y + YZ Redundant by consensus theorem p.40 #17 0 = 0 + XZ + X’Y = XZ + X’Y

  11. EXAMPLE: CONVERT to SOP FORM (p. 51) (A+B+C’) (A+B+D) (A+B+E) (A+D’+E)(A’+C) Let X=A+B, Y=C’, Z=D (X+Y)(X+Z) = X+YZ (#8D p.40) =(A+B+C’D) (A+B+E) (A+D’+E) (A’+C) Let X=A, Y=D’+E, Z=C,  (X+Y)(X’+Z) = XZ+X’Y (#16 p.40) =(A+B+C’D)(A+B+E) (AC+A’(D’+E)) =(A+B+C’D)(A+B+E) (AC+A’D’+A’E)) by distr. law Let X=A+B, Y=C’D, Z=E,  (X+Y)(X+Z) = X+YZ =(A+B+C’DE) (AC+A’D’+A’E)) Mult. out by distr. law and eliminate terms such asAA’D’=AAC+AA’D’+AA’E+ABC+A’BD’+A’BE +C’DEAC+C’DEA’D+C’DEA’E = AC + ABC + A’BD + A’BE + A’C’DE Let X=AC, Y=B X+XY = X (#10 p.40) = AC + A’BD’ + A’BE + A’C’DE

  12. EXAMPLE: CONVERT to POS FORM (p. 51) AC + A’BD’ + A’BE + A’C’DE (A’ is common to three terms) = AC + A’(BD’ + BE + C’DE) by distr. law Let X=A,X’=A’, Y=BD’ + BE + C’DE, Z=CXZ+X’Y=(X+Y)(X’+Z) =(A+BD’+BE+C’DE)(A’+C)=(A+C’DE+BD’+BE)(A’+C) -re-arranging = A+C’DE +B(D’+E ) (A’+C) by distr. law Let X= A+C’DE, Y=B, Z=D’+E X+YZ=(X+Y)(X+Z) (8D) = (A+B+C’DE)(A+C’DE +D’+E)(A’+C) But E+C’DE = E (using X+XY=X) so C’DE is redundant = (A+B+C’DE) (A+D’+E)(A’+C) Let X=A+B, Y=C’, Z=DEX+YZ=(X+Y)(X+Z) =(A+B+C’)(A+B+DE) (A+D’+E)(A’+C) Let X=A+B, Y=D, Z=EX+YZ=(X+Y)(X+Z) =(A+B+C’) (A+B+D)(A+B+E) (A+D’+E)(A’+C)

  13. A F = A  B B EXCLUSIVE-OR FUNCTION One last gate type is the XOR Gate (Exclusive OR gate). XOR A B Y0 0 00 1 11 0 11 1 0 XOR gate is common in logic circuits that do binary addition/subtraction. Note that:F = A  B= A’B + AB’ = (A+B) (AB)’ -- see eqns. 3-17 to 3-24 p. 52 for XOR theorems

  14. EQUIVALENCE FUNCTION (º) Equivalence is the complement of XOR. º A F = (A  B)’= A º B A B Y0 0 10 1 01 0 01 1 1 B º A º B A -- alternate symbol B Note: In order to simplify expressions containing AND and OR as well as XOR and º, it is usually desirable to eliminate XOR and º by subst. defns. in terms of AND and OR.

  15. POSITIVE and NEGATIVE LOGIC • General Concept • Positive Logic • High Voltage => Logic 1 • Low Voltage => Logic 0 • Negative Logic • High Voltage => Logic 0 • Low Voltage => Logic 1

  16. POSITIVE and NEGATIVE LOGIC (cont’d) • Implication • Positive Logic High Voltage => Logic 1 Low Voltage => Logic 0 Voltage A B F Low Low Low Low High Low High Low Low High High High Logic A B F 0 0 0 0 1 0 1 0 0 1 1 1 - Equivalent gate: AND

  17. POSITIVE and NEGATIVE LOGIC (cont’d 2) • Implication • Negative Logic High Voltage => Logic 0 Low Voltage => Logic 1 Voltage A B F Low Low Low Low High Low High Low Low High High High Logic A B F 1 1 1 1 0 1 0 1 1 0 0 0 - Equivalent gate: OR

  18. POSITIVE and NEGATIVE LOGIC (cont’d 3) • Implication • Positive Logic High Voltage => Logic 1 Low Voltage => Logic 0 Voltage A B F Low Low Low Low High High High Low High High High High Logic A B F 0 0 0 0 1 1 1 0 1 1 1 1 - Equivalent gate: OR

  19. POSITIVE and NEGATIVE LOGIC (cont’d 4) • Implication • Negative Logic High Voltage => Logic 0 Low Voltage => Logic 1 Voltage A B F Low Low Low Low High High High Low High High High High Logic A B F 1 1 1 1 0 0 0 1 0 0 0 0 - Equivalent gate: AND -- Negative logic Thm. p. 56 (skip proof)

  20. What do you need to know? • De Morgan’s Laws • Duality (covered in previous presentation) • NAND, NOR gates • Multiplying Out and Factoring Expressions • XOR Gate, Equivalence Gate • Positive and Negative Logic

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