Chapter No.6: Boolean Algebra
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By Sir Moin
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Boolean Algebra Rule # 10
Rule 10: A + A.B = A
Proof:
This rule can be proved by applying the distributive law, rule 2 and rule 4 as follows.
| A + B | = A(1 + B) | Factoring A (Distributive law) |
| = A.1 | Rule 2: B + 1 = 1 |
| = A | Rule 4: A.1= A |
The proof is shown in the Truth table.
| A | B | A.B | A + A.B |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
|-------------------------------------Equal-----------------------------------|
Boolean Algebra Rule # 11
Rule 11: A + A.B = A + B
Proof:
This rule can be proved as.
| A + A.B | = (A + A.B) + A.B | Rule 10: A = A + A.B |
| | = (AA + AB) + AB | Rule 6: A = AA |
| = AA + AB + AA + AB | Rule 8: Adding AA = 0 |
| | = (A + A) (A + B) | Factoring |
| | = 1 . (A + B) | Rule 7: A + A= 1 |
| | = (A + B) | Rule 4: Drop 1 |
The proof is shown in the Truth table.
| A | B | AB | A + AB | A + B |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
|--------Equal-----|
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