1. CEC 220 Digital Circuit Design Boolean Algebra Friday, January 17 CEC 220 Digital Circuit Design Slide 1 of 22

2. Lecture Outline Friday, January 17 CEC 220 Digital Circuit Design Introduction Basic Operations: NOT, AND, OR Representations of Boolean Expressions Basic Boolean Theorems Implementation of Boolean Expressions Slide 2 of 22

3. Boolean Algebra Introduction Friday, January 17 CEC 220 Digital Circuit Design 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 Slide 3 of 22

4. Boolean Algebra Boolean Logic Friday, January 17 CEC 220 Digital Circuit Design Two logic levels  TRUE = HIGH = 1  FALSE = LOW = 0 +5 V 0 101 Time Transition from logic 1 to logic 0 does not take place instantaneously in real digital systems +5 V 0 Logic 1 Logic 0 Intermediate values may be visible for an instant Slide 4 of 22

5. Boolean Algebra Basic Operations: NOT, AND, OR Friday, January 17 CEC 220 Digital Circuit Design AC AC 01 10 AC FT TF Slide 5 of 22

6. Boolean Algebra Basic Operations: NOT, AND, OR Friday, January 17 CEC 220 Digital Circuit Design Logical AND  Description: o 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 ABC 000 010 100 111 ABC FFF FTF TFF TTT Slide 6 of 22

7. Boolean Algebra Basic Operations: NOT, AND, OR Friday, January 17 CEC 220 Digital Circuit Design Logical OR  Description: o 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 ABC 000 011 101 111 ABC FFF FTT TFT TTT Slide 7 of 22

8. Boolean Algebra Logic Gates and Boolean Expressions Friday, January 17 CEC 220 Digital Circuit Design 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). B’ A  B’ A  B’+C Slide 8 of 22

9. Boolean Algebra Logic Gates and Boolean Expressions Friday, January 17 CEC 220 Digital Circuit Design C+D A(C+D) BE Slide 9 of 22

10. Boolean Algebra Logic Gates and Boolean Expressions Friday, January 17 CEC 220 Digital Circuit Design 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’ Slide 10 of 22

11. Boolean Algebra Truth Tables of a Logic Circuit Friday, January 17 CEC 220 Digital Circuit Design Determine the truth table for the output (F) of the logic circuit AB 00 01 10 11 A’ 1 1 0 0 F=A’+B 1 1 0 1 Two variables  Four possible inputs (i.e. 2 n ) Slide 11 of 22

12. A+C 0 1 0 1 1 1 1 1 Boolean Algebra Truth Tables of a Logic Circuit Friday, January 17 CEC 220 Digital Circuit Design Determine the truth table for the output of the logic circuit ABC 000 001 010 011 100 101 110 111 B’+C 1 1 0 1 1 1 0 1 A+CB’+C(A+C)(B’+C) 0 1 0 1 1 1 0 1 B’ 1 1 0 0 1 1 0 0 Slide 12 of 22

13. Boolean Algebra Basic Boolean Theorems Friday, January 17 CEC 220 Digital Circuit Design Basic Theorems  Principle of DUALITY: o Given any Boolean expression its DUAL expression can be obtained by: – Replace “ “ by “ + “ (and vice versa), also – Replace “ 0 “ by “ 1 “ (and vice versa) Slide 13 of 22

14. Boolean Algebra Basic Boolean Theorems Friday, January 17 CEC 220 Digital Circuit Design Basic Theorems  Operations with 0 and 1 x  0 = 0 x  1 = x x 0 x0x0 000 100 x 1 x1x1 010 111 x +0 = x x +1 = 1 x 0 x +0 000 101 x 1 x +1 011 111 Expression Dual of Expression Slide 14 of 22

15. Boolean Algebra Basic Boolean Theorems Friday, January 17 CEC 220 Digital Circuit Design Idempotent Law: Laws of Complementarity Involution Law x + x = x x  x = x Expression Dual of Expression Expression Dual of Expression Slide 15 of 22

16. Boolean Algebra More Boolean Theorems Friday, January 17 CEC 220 Digital Circuit Design Commutative Law Associative Law Distributive Law x  y = y  x Expression Dual of Expression x + y = y + x (x  y)  z = x  (y  z) Expression Dual of Expression (x + y) + z = x + (y + z) x  (y + z) = ( x  y) + (x  z) Expression Dual of Expression x + (y  z) = ( x + y)  (x + z) Slide 16 of 22

17. Boolean Algebra More Boolean Theorems Friday, January 17 CEC 220 Digital Circuit Design Let’s verify the Distributive Law via a truth table y + z 0 1 1 1 0 1 1 1 xyz 000 001 010 011 100 101 110 111 x (y + z) = x y + x z x(y + z) 0 0 0 0 0 1 1 1 LHS x y 0 0 0 0 0 0 1 1 xyz 000 001 010 011 100 101 110 111 x z 0 0 0 0 0 1 0 1 RHS x y + x z 0 0 0 0 0 1 1 1 y + zx(y + z) x yx z x y + x z Slide 17 of 22

18. Boolean Algebra Boolean Algebra Examples Friday, January 17 CEC 220 Digital Circuit Design 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 Slide 18 of 22

19. Boolean Algebra Boolean Algebra Examples Friday, January 17 CEC 220 Digital Circuit Design Examples: LHS Last example: X+XZ = X Distributive Law (Dual) Idempotent Law Distributive Law Last example: X+XY = X Slide 19 of 22

20. Boolean Algebra A Circuit Example Friday, January 17 CEC 220 Digital Circuit Design Determine the Output of the Following Circuit Design a Simpler Circuit with the Same Output x + (y  z) = ( x + y)  (x + z) Distributive Law (Dual) Slide 20 of 22

21. Boolean Algebra An Inverter Friday, January 17 CEC 220 Digital Circuit Design Implementation of an inverter (Lab 1) A simple RTL logic inverter 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 Vin Low Vin High Slide 21 of 22

22. Next Lecture Friday, January 17 CEC 220 Digital Circuit Design DeMorgan’s Laws Simplification Theorems Slide 22 of 22