The most obvious way to simplify Boolean expressions is to manipulate them in the same way as normal algebraic expressions are manipulated. By combining the laws of Boolean algebra and our knowledge of logic gates we form several useful rules that may be used in manipulating and simplifying Boolean algebra expressions.
A + B = B + A
Rules 1-9, as listed in the table , are the core precepts from which rules 10-12 are derived. Note that in each case, A, B, or C can either represent a single variable of a combination of variables.
Rule Number | Boolean Expression |
1 |
|
2 |
|
3 |
|
4 |
|
5 |
|
6 |
|
7 |
|
8 |
|
9 |
|
10 |
|
11 |
|
12 |
|
Table 1-10 Rules of Boolean Algebra
All these rules, in particular rules 1-9, are easily verified using truth tables.
Let us examine two methods by which we can prove the relationships of rules 10-12. First we use the laws and rules of Boolean algebra. Second we employ the use of truth tables.
| A + AB = A (1+B) | distributive law |
= A .1 | rule 2 |
| =A | rule 4 |
Method 2:
A
B
AB
A + AB
0
0
0
0
0
1
0
0
1
0
0
1
1
1
1
1
Table illustrating A+AB=A
As the shaded columns are equal then the rule has be shown to be correct.
Similarly for Rule 12, we can apply the same two methods to prove the relationship.
Method 1:
| (A + B)(A+C) = AA +AC + BA + BC | distributive law |
| = A + AC + BA + BC | rule 7 |
| = A (1 + C) + BA + BC | distributive law |
| = A.1 +BA + BC | rule 2 |
| = A + BA + BC | rule 4 |
| = A (1 + B) + BC | distributive law |
| = A.1 + BC | rule 2 |
| = A + BC | rule 4 |
Method 2:
A | B | C | A + B | A+C | (A + B)(A+C) | BC | A + BC |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table illustrating (A + B)(A+C) = A+BC
Again, as the shaded columns are equal then the rule has be shown to be correct
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