CEC 220 Digital Circuit Design Boolean Algebra

1. CEC 220 Digital Circuit DesignBoolean Algebra CEC 220 Digital Circuit Design

2. Lecture Outline • Introduction • Basic Operations: NOT, AND, OR • Representations of Boolean Expressions • Basic Boolean Theorems • Implementation of Boolean Expressions CEC 220 Digital Circuit Design

3. Boolean AlgebraIntroduction • Boolean Algebra • In 1849 George Boole published a scheme for the algebraic description of logic processes • In the 1930’s Claude Shannon used Boolean Algebra to describe circuits built with switches • Boolean Algebra is an effective tool for describing logic circuits CEC 220 Digital Circuit Design

4. Boolean AlgebraBoolean Logic T ime • Two logic levels • TRUE = HIGH = 1 • FALSE = LOW = 0 +5 1 0 1 V 0 +5 Transition from logic 1 to logic 0 does not take place instantaneously in real digital systems Logic 1 V Intermediate values may be visible for an instant Logic 0 0 Time CEC 220 Digital Circuit Design

5. Boolean AlgebraBasic Operations: NOT, AND, OR • Logical NOT • Description: • The output is the complement/inverse/opposite of the input • Symbolic Representation (NOT gate): • Truth Table Representation: • Boolean Description: C = NOT A A C or C = or C = A’ CEC 220 Digital Circuit Design

6. Boolean AlgebraBasic Operations: NOT, AND, OR • Logical AND • Description: • The output is TRUE if and only if all the inputs are TRUE • Symbolic Representation (AND gate): • Truth Table Representation: • Boolean Description: C = A AND B or C = A B or C = AB CEC 220 Digital Circuit Design

7. Boolean AlgebraBasic Operations: NOT, AND, OR • Logical OR • Description: • The output is TRUE if any of the inputs are TRUE • Symbolic Representation (OR gate): • Truth Table Representation: • Boolean Description: C = A OR B • or C = A + B CEC 220 Digital Circuit Design

8. Boolean AlgebraLogic Gates and Boolean Expressions • Derive an expression for the output of this logic circuit? • Eventually we will omit the “” in the AND gate and “+” in the OR gate • The logic expression is a function of three variables (A, B, and C). AB’ AB’+C B’ CEC 220 Digital Circuit Design

9. Boolean AlgebraLogic Gates and Boolean Expressions • Derive an expression for the output of this logic circuit? • The output is which may also be written as [A(C+D)]’+BE • This expression has five variables (A, B, C, D, and E) C+D A(C+D) + BE CEC 220 Digital Circuit Design

10. Boolean AlgebraLogic Gates and Boolean Expressions • Literals • Each appearance of a variable or its complement in an expression is referred to as a literal. • Example: • The expression has three variables (A, B, and C) • The expression has 10 literals AB’C+A’B+A’BC’+B’C’ CEC 220 Digital Circuit Design

11. Boolean AlgebraTruth Tables of a Logic Circuit • Determine the truth table for the output (F) of the logic circuit Two variables Four possible inputs (i.e. 2n) CEC 220 Digital Circuit Design

12. Boolean AlgebraTruth Tables of a Logic Circuit • Determine the truth table for the output of the logic circuit CEC 220 Digital Circuit Design

13. Boolean AlgebraBasic Boolean Theorems • Basic Theorems • Principle of DUALITY: • Given any Boolean expression its DUAL expression can be obtained by: • Replace “ • “ by “ + “ (and vice versa), also • Replace “ 0 “ by “ 1 “ (and vice versa) CEC 220 Digital Circuit Design

14. Boolean AlgebraBasic Boolean Theorems • Basic Theorems • Operations with 0 and 1 Expression Dual of Expression x0 = 0 x+1 = 1 x1 = x x+0 = x CEC 220 Digital Circuit Design

15. Boolean AlgebraBasic Boolean Theorems • Idempotent Law: • Laws of Complementarity • Involution Law Expression Dual of Expression x x = x x + x = x Expression Dual of Expression = 0 = 1 CEC 220 Digital Circuit Design

16. Boolean AlgebraMore Boolean Theorems • Commutative Law • Associative Law • Distributive Law Expression Dual of Expression x  y = y  x x + y = y + x Expression Dual of Expression (x + y) + z = x + (y + z) (x y)  z = x (y z) Expression Dual of Expression x  (y + z) = (x y) + (x  z) x + (y  z) = (x + y)  (x + z) CEC 220 Digital Circuit Design

17. Boolean AlgebraMore Boolean Theorems • Let’s verify the Distributive Law via a truth table x (y + z) = x y + x z LHS RHS CEC 220 Digital Circuit Design

18. Boolean AlgebraBoolean Algebra Examples • Examples • Prove the following algebraically Distributive Law LHS Complementarity Law • Operations with 0 and 1 LHS • Operations with 0 and 1 Distributive Law • Operations with 0 and 1 CEC 220 Digital Circuit Design

19. Boolean AlgebraBoolean Algebra Examples • Examples: Distributive Law (Dual) LHS Distributive Law • Idempotent Law • Last example: X+XZ = X • Last example: X+XY = X OR CEC 220 Digital Circuit Design

20. Boolean AlgebraA Circuit Example • Determine the Output of the Following Circuit • Design a Simpler Circuit with the Same Output Distributive Law (Dual) x + (y  z) = (x + y)  (x + z) CEC 220 Digital Circuit Design

21. Boolean AlgebraAn Inverter • Implementation of an inverter A simple RTL logic inverter Vin High Vin Low • Interpret voltages per the TTL standard: • 0 to 0.8 volts = Boolean 0 (Low) • 2.2 to 5.0 volts = Boolean 1 (High) Vout = NOT Vin CEC 220 Digital Circuit Design

22. Next Lecture • DeMorgan’s Laws • Simplification Theorems CEC 220 Digital Circuit Design