Laws & Rules of Boolean Algebra
Commutative law of addition
Commutative law of addition,
A+B = B+A
the order of ORing does not matter.
Commutative law of multiplication
Commutative law of multiplication
AB = BA
The order of ANDing does not matter.
Associative law of addition
Associative law of addition
A+(B + C) = (A + B) + C
The grouping of ORed variables does not matter
Associative law of multiplication
Associative law of multiplication
A(BC) = (AB)C
The grouping of ANDed variables does not matter
Distributive Law
A(B + C) = AB + AC
Boolean Rules
1) A + 0 = A
• In math if you add 0 you have changed nothing
• Im Boolean Algebra ORing with 0 changes
nothing
Boolean Rules
2) A + 1 = 1
• ORing with 1 must give a 1 since if any input is
1 an OR gate will give a 1
Boolean Rules
3) A•0 = 0
• In math if 0 is multiplied with anything you get
get 0. If you AND anything with 0 you get 0
4) A • 1 = A
• ANDing anything with 1 will yield the anything
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
Proof :
A + AB = A(1 +B) DISTRIBUTIVE LAW
= A•1 RULE 2: (1+B) =1
= A RULE 4: A•1 = A
11) A + AB = A + B
• 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
END OF BOOLEAN RULES & LAWS
Sumber Berita :
https://onlinelearning.uhamka.ac.id
