1. BOOLEAN ALGEBRA Only involves in calculations of TRUE and FALSE; either be inputs or output. When a logic statement is TRUE it is assigned a Boolean logic value of one (1). When a logic statement is FALSE it is assigned a Boolean logic value of zero (0)

2. TWO VARIABLES Let have two binary variables : A and B. Each of the variables can have either TRUE (1) or FALSE (0) value. CombinationAB Combination 1FALSE (0) Combination 2FALSE (0)TRUE (1) Combination 3TRUE (1)FALSE (0) Combination 4TRUE (1)

3. There are only four ways to combine these two variables, i.e.; ABCombination 0000 0101 1010 1111

4. OR and AND In this examples, there are two inputs, i.e., A and B, and output, i.e.,C. OR symbol A A + B = C (Probability add) B C AND symbol A C A x B = C (Probability multiply) B or A.B = C or AB = C

5. Remember there are four ways to combine the two outputs ; OR means either. If any of the values is TRUE (1) the answer (resulting value) is TRUE (1). AND meant both. Only if both A and B are true, only then the resulting value is TRUE (1). AB 00 01 10 11 ABC = A + BD = AB 0000 0110 1010 1111

6. For OR operation, there is only one out of four chance to be FALSE (0), i.e., when both A and B are FALSE. On the other hand for AND operation, there is only one chance out of four to TRUE (1), i.e., when both A and B are TRUE. What happen if more two input variables involved, say X, Y, Z. Let; C = X + Y + Z D = X.Y.Z

7. XYZC = X+Y+ZD = X.Y.Z 00000 00110 01010 01110 10010 10110 11010 11111

8. Z Y X Z Y C = X + Y +Z OR gate AND gate X

9. Those qualities which are valuable tools for working problems are designated as Theorems. Intermediate results necessary in the proof of the theorems and results, which are included only for the sake of completeness, are called Lemmas. LEMMA AND THEOREMS OF BOOLEAN ALGEBRA

10. LEMMA 1: The 0 and 1 are unique. LEMMA 2: For every element a in K, a+a = a, and a.a = a LEMMA 3: For every element a in K, a+1 = 1, a.0 = 0 LEMMA 4: The element 1 and 0 are distinct LEMMA 5: For every pair of elements of a and b in K, a+ab = a and a (a+b) = a LEMMA 6: The for every a in K is unique, a.a = 0 and a +a = 1 LEMMA 7: For every a in K, a = a LEMMA 8 : a [(a + b) + c] = [(a + b) + c] a = a THEOREM 9: For any three elements a, b and c in K, a+ (b+c) = (a+b) + c and a(bc) = (ab)c THEOREM 10: For any pair of elements a and b in K, a+ab = a+b; a(a+b) = ab THEOREM 11: For any pair of elements a and b in K, a+b = ab; ab= a+ab