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

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

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

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

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

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.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

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

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

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

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

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

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

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

1 Example 1: Signal Pushing INPUTOUTPUT ABW Apply all input combinations, follow the logic through the circuit and complete the truth table Animated Boolean 1.15

Example 2: Signal Pushing INPUTOUTPUT ABCY Boolean 1.16

Exercise 1: Signal Pushing Apply all input combinations and complete the truth table. INPUTOUTPUT ABCW Boolean 1.17

Exercise 2: Signal Pushing INPUTOUTPUT ABCDZ Boolean 1.18

Sum-of-Products (SOP) Boolean 1.19

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

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

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

Example 2: Sum of Products (ABC)+(ABC)+(ABC)+(ABC) = W INPUTOUTPUT ABCW Boolean 1.23

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

©Paul R. Godin prgodin gmail.com END Boolean 1.25