RULES OF BOOLEAN ALGEBRA

1. RULES OF BOOLEAN ALGEBRA

2. BASIC RULES OF BOOLEAN ALGERBA

3. De Morgan’s Theorems • It states, that the complement of any expression can be obtained by replacing each variable and element with its complement and changing OR operators (+) with AND operators(.) and AND operator (.) with OR operators (+). • These theorems can be expressed as follows: • (A+B)’=(A)’.(B)’ First De Morgan’s Theorem. • (A.B)’=(A)’+(B)’ Second De Morgan’s Theorem. De Morgan’s theorems can also applicable to expressions in which there are more than two variables.

4. Three steps of simplification (demorganization) • Complement each of the individual variables. • Change all the OR’s (+) to AND (.) and AND’s(.) to OR’s(+). • Complement the whole Boolean function.

5. Examples 1 • Demorganize the following expression: AB’ + A’B’C +AC Step No. 1: Complement each of the individual variables Means : AB’ + A’B’C + AC  A’B’’ + A’’B’’C’ + A’C’ Step No. 2: Change all the OR’s (+) to AND (.) and AND’s(.) to OR’s(+)

6. Examples 1 Means: A’B’’ + A’’B’’C’ + A’C’ A’B + ABC’ + A’C’ A’B + ABC’ + A’C’  (A’+B ).(A+B+C’).(A’+C’) Step No. 3: Complement the whole Boolean function Means: (A’+B ).(A+B+C’).(A’+C’) ((A’+B ).(A+B+C’).(A’+C’))’

7. Truth Table For This Example 1. Before Demorganization. AB’ + A’B’C +AC

8. Truth Table For This Example 2. After Demorganization. ((A’+B ).(A+B+C’).(A’+C’))’

9. Truth Table For This Example • Now Compare the results of the expression before and after Demorganization The Results are identical in both the tables. Before Demorganization After Demorganization

10. Examples 1

11. Truth Table For This Example 1. Before Demorganization. (A+B’).(A+B+C’).(AB)

12. Truth Table For This Example 2. After Demorganization. (A’.B+A’.B’.C+A’+B’)’

13. Truth Table For This Example • Now Compare the results of the expression before and after Demorganization The Results are identical in both the tables. Before Demorganization After Demorganization