Laws and Rules of Boolean Algebra
- Commutative law of addition
AB = BA
the order of ANDing does not matter
- Commutative law of Multiplication
the order of ORing does not matter
- Associative law of addition
A + (B + C) = (A + B) + C
The grouping of ORed variables does not matter
- Associative law of multiplication
A(BC) = (AB)C
The grouping of ANDed variables does not matter
- Distributive Law
A(B + C) = AB + AC
(A+B)(C+D) = AC + AD + BC + BD
Boolean Rules
1) A + 0 = A
-In math if you add 0 you have changed nothing
-In Boolean Algebra ORing with 0 changes nothing
2) A + 1 = 1
-ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1
3) A • 0 = 0
-In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0
4) A • 1 = A
5) A + A = A
-ORing with itself will give the same result
6) A + A = 1
-Either A or A must be 1 so A + A =1
7) A • A = A
-ANDing with itself will give the same result
8) A • A = 0
-In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0
9) A = A
-If you not something twice you are back to the beginning
10) A + AB = A
Proof:
A + AB = A(1 +B) DISTRIBUTIVE LAW
= A·1 RULE 2: (1+B)=1
= A RULE 4: A·1 = A
-If A is 1 the output is 1 , If A is 0 the output is B
Proof:
A + AB = (A + AB) + AB RULE 10
= (AA +AB) + AB RULE 7
= AA + AB + AA +AB RULE 8
= (A + A)(A + B) FACTORING
= 1·(A + B) RULE 6
= A + B RULE 4
12) (A + B)(A + C) = A + BC
PROOF
(A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW
= A + AC + AB + BC RULE 7
= A(1 + C) +AB + BC FACTORING
= A.1 + AB + BC RULE 2
= A(1 + B) + BC FACTORING
= A.1 + BC RULE 2
= A + BC RULE 4
Nama:Yogga Tolly Dewanto
Nim :2003015136
Kelas:2F
sumber berita: https//onlinelearning.uhamka.ac.id
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