1. Boolean Logic 1 Technician Series Boolean 1.1 ©Paul Godin Created Jan 2015 prgodin @ gmail.com

2. Boolean Simplification ◊Boolean equations are used to describe a logic circuit’s function. ◊Equations can become complex and require simplification. ◊There are laws and theorems to help simplify complex Boolean problems. ◊Manipulating Boolean equations follows many of the rules of standard algebra. Boolean 1.2

3. The 3 Boolean Laws ◊Commutative: ◊Addition: A + B = B + A ◊Multiplication: AB = BA ◊Associative: ◊Addition: A + (B + C) = (A + B) + C ◊Multiplication: A(BC) = (AB)C ◊Distributive: ◊A(B + C) = AB + AC ◊(A + B)(C + D) = AC + AD + BC + BD Boolean 1.3

4. The 10 Basic Rules (part 1) 1.Anything ANDed with a 0 is equal to 0:A ● 0 = 0 2.Anything ANDed with a 1 is equal to itself:A ● 1 = A 3.Anything ORed with a 0 is equal to itself:A + 0 = A 4.Anything ORed with a 1 is equal to 1:A + 1 = 1 5.Anything ANDed with itself is equal to itself:A ● A = A Boolean 1.4

5. The 10 Basic Rules (part 2) 6.Anything ORed with itself is equal to itself:A + A = A 7.Anything ANDed with its own complement equals 0:A ● A = 0 8.Anything ORed with its own complement equals 1:A + A = 1 9.Anything complemented twice is equal to the original: A = A 10.The two variable rules: a) A + AB = A + B b) A + AB = A + B c) A + AB = A Boolean 1.5

6. De Morgan’s Theorem Review ◊De Morgan’s Theorem allows the inversion of an expression to be broken up into inversions of individual variables. ◊Inversion of an expression: A + B ◊Inversion of individual variables: A ● B “Break the bar and change the sign” Boolean 1.6

7. 7.A ● 0 = ___ 8.A + A = ____ 9.A + A = ____ 10.A + AB = ____ 11.A ● 1 = ____ 12.A = ____ Basic Boolean Rules Exercise 1 1.A + 0 = ____ 2.A + AB = ____ 3.A + 1 = ____ 4.A + AB = ____ 5.A ● A = ____ 6.A ● A = ____ Determine the outcome of the following: Boolean 1.7

8. Basic Boolean Rules Exercise 2 Determine the output of the following gates ? 0 A ? 1 ? 0 ? 1 A A A’ A A Boolean 1.8

9. Boolean Simplification ◊Boolean equations can be simplified using algebraic methods, using the Boolean rules and laws to reduce the equation. Boolean 1.9

10. Examples of Boolean Reduction 1 ◊Consider CD(D+DF) ◊CD(D) Rule 10a where D+DF=D ◊CDD Associative Law ◊CD Rule 5 where DD=D ◊Consider C’D’(C+D)’ ◊C’D’(C’D’)DeMorgan where (C+D)=(CD) ◊C’C’D’D’Associative where brackets removed ◊C’D’Rule 5 where C’C’=C’ and D’D’=D’ Boolean 1.10

11. Examples of Boolean Reduction 2 ◊Consider (C+D)(C+D’) ◊CC+CD’+DC+DD’ Distributive ◊C+CD’+CD+0Rule 5: CC=C; Rule 7:DD’=0, ◊C+CD’+CDRule 3: (A+0=A) ◊(C+CD’)+CD Associative ◊C+CDRule 10c: C+CD’=C ◊C+CRule 10c: C+CD=C ◊CRule 6: C+C=C ◊Consider C’+CDE+E ◊(C’+CDE)+EAssociative ◊C’+DE+ERule 10b: C’+CDE=C’+DE ◊C’+(E+DE)Associative, Commutative ◊C’+ERule 10c: E+DE=E Boolean 1.11

12. Exercise 3 ◊Simplify the following: ◊ A’+AB’+B ◊A+A’B+B’C+AC ◊AB’+A’CD+B+C’+D’ Other examples may be given in class Boolean 1.12

13. Pushing a Signal ◊Signal Pushing is a technique of applying an input value and following its progression through a circuit. ◊This method is used extensively when analysing and troubleshooting circuits. Boolean 1.13

14. Truth Tables and Signal Pushing ◊A truth table can be derived from a circuit by signal pushing. ◊All possible input combinations are applied to determine the output of the circuit. Boolean 1.14

15. 1 Example 1: Signal Pushing INPUTOUTPUT ABW 00 01 10 11 Apply all input combinations, follow the logic through the circuit and complete the truth table. 0 1 0 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 0 0 Animated Boolean 1.15

16. Example 2: Signal Pushing INPUTOUTPUT ABCY 0000 0010 0100 0111 1000 1011 1100 1111 Boolean 1.16

17. Exercise 1: Signal Pushing Apply all input combinations and complete the truth table. INPUTOUTPUT ABCW 000 001 010 011 100 101 110 111 Boolean 1.17

18. Exercise 2: Signal Pushing INPUTOUTPUT ABCDZ 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Boolean 1.18

19. Sum-of-Products (SOP) Boolean 1.19

20. Boolean from the Truth Table ◊A Boolean equation derived from the truth table takes on a “Sum-of-Products” form. ◊Sum-of-Products are ANDed statements (products) that are ORed together (sum). Example: (A●B)+(C●D) ◊From the Truth Table, for all output values that equal “1”, the ANDed input values are written. ◊If the input value is “0”, the complimented input is indicated. Boolean 1.20

21. SOP from the Truth Table This example demonstrates how the S.O.P. equation is determined. INPUTOUTPUT ABW 001 010 101 110 (A●B)+(A●B)=W Boolean 1.21

22. S.O.P. Simplification ◊Once the Boolean equation in S.O.P. form is determined, standard simplification rules are applied. ◊Example: (A●B)+(A●B)=W B(A+A)=W B(1)=W B=W Boolean 1.22

23. Example 2: Sum of Products (ABC)+(ABC)+(ABC)+(ABC) = W INPUTOUTPUT ABCW 0000 0010 0100 0111 1000 1011 1101 1111 Boolean 1.23

24. 3-Boolean Expression Simplified (ABC)+(ABC)+(ABC)+(ABC) = W (ABC)+(ABC)+(ABC)+(ABC)+(ABC)+(ABC) = W BC(A+A)+AB(C+C)+AC(B+B) = W BC+AB+AC = W Boolean 1.24

25. ©Paul R. Godin prgodin ° @ gmail.com END Boolean 1.25