1. CEC 220 Digital Circuit Design Boolean Algebra II Fri, Sept 4 CEC 220 Digital Circuit Design Slide 1 of 13

2. Lecture Outline Fri, Sept 4 CEC 220 Digital Circuit Design Basic Boolean Theorems Implementation of Boolean Expressions Slide 2 of 13

3. Boolean Algebra Review of AND, OR, NOT gates Fri, Sept 4 CEC 220 Digital Circuit Design The following signals were observed in the Lab  The inputs to the box are A and B  What type of gate is in the box? o AND, OR, NOT ? ? A B A B ? A B ? A B Slide 3 of 13

4. Boolean Algebra Basic Boolean Theorems Fri, Sept 4 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 4 of 13

5. Boolean Algebra Basic Boolean Theorems Fri, Sept 4 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 5 of 13

6. Boolean Algebra Basic Boolean Theorems Fri, Sept 4 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 6 of 13

7. Boolean Algebra More Boolean Theorems Fri, Sept 4 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 7 of 13

8. Boolean Algebra More Boolean Theorems Fri, Sept 4 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 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 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 xyz 000 001 010 011 100 101 110 111 xyz 000 001 010 011 100 101 110 111 Slide 8 of 13

9. Boolean Algebra Boolean Algebra Examples Fri, Sept 4 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 9 of 13

10. Boolean Algebra Boolean Algebra Examples Fri, Sept 4 CEC 220 Digital Circuit Design Examples: LHS Last example: X+XZ = X Distributive Law (Dual) Idempotent Law Distributive Law Last example: X+XY = X OR Slide 10 of 13

11. Boolean Algebra A Circuit Example Fri, Sept 4 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 11 of 13

12. Boolean Algebra An Inverter Fri, Sept 4 CEC 220 Digital Circuit Design Implementation of an inverter 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 12 of 13

13. Next Lecture Fri, Sept 4 CEC 220 Digital Circuit Design DeMorgan’s Laws Simplification Theorems Slide 13 of 13