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Simplifying Boolean Expressions Using Logic Gates and DeMorgan's Theorems

This chapter dives into the simplification of Boolean expressions through the application of logic gates and DeMorgan's Theorems. It covers how to factor out common variables in expressions, apply theorems for simplifying complex digital logic, and provides examples such as ACD + ABCD and its transformation using fundamental algebraic identities. The text illustrates the importance of maintaining parentheses during simplification and discusses alternate gate representations, including NAND and NOR gates, showcasing their universality in digital circuits.

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Simplifying Boolean Expressions Using Logic Gates and DeMorgan's Theorems

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Presentation Transcript

  1. Chapter 3 Logic Gates & Boolean Algebra

  2. Algebra Example Simplify this expression: ACD + ABCD Factor out common variables: CD(A + AB) Apply theorem 15a: CD(A + B) or ACD + BCD

  3. DeMorgan’s Theorems Demorgan’s Theorems allow us to break up a “bar”which is over an entire expression: (X + Y) = X ● Y (X ● Y) = X + Y

  4. DeMorgan’s Theorem Example (A B + C) Simplify this expression: Apply Demorgan’s: (A B) ●C Apply Demorgan’s: (A + B) ●C (notice importance of keeping the parentheses!) Cancel double bars: (A + B) ●C or AC + BC

  5. DeMorgan’s Theorem Example (A + C) ( B + D) Simplify this expression: Apply Demorgan’s: (A + C) + (B + D) Apply Demorgan’s: (A ●C) + (B ● D) Cancel double bars: AC + BD

  6. DeMorgan’s Theorem Example (A + BC) (D + EF) Simplify this expression: Apply Demorgan’s: (A + BC) + (D + EF) Apply Demorgan’s: (A ● BC) + (D ● EF) Apply Demorgan’s: (A ● (B+C)) + (D ● (E+F)) Distribute: AB + AC + DE + DF

  7. Alternate Gate Representations (X + Y) = X ● Y X Y X + Y X Y X Y X ● Y = X + Y

  8. Alternate Gate Representations NOR Gates NAND Gates

  9. NAND Gates are Universal Gates X X ● X = X X X X Y XY X ● Y XY X Y X Y X + Y

  10. NAND Gates are Universal Gates X Y X ● Y = X + Y = X + Y X + Y X Y X Y X ● Y X Y

  11. Using NAND Gates to Simplify 2 CHIPS: 7408 = AND 7432 = OR

  12. Using NAND Gates to Simplify

  13. Using NAND Gates to Simplify 1 CHIP: 7400 = NAND

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