Tugas 4 Rangkuman Materi Aljabar Boolean

 Laws & Rules of Boolean Algebra 

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

(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 


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 0. If you AND anything with 0 you get 0

Boolean Rules 4) A • 1 = A  ANDing anything with 1 will yield the anything

Boolean Rules 5) A + A = A  ORing with itself will give the same result

Boolean Rules 6) A + A = 1  Either A or A must be 1 so A + A =1

Boolean Rules 7) A • A = A  ANDing with itself will give the same result

Boolean Rules 8) A • A = 0  In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.
Boolean Rules 9) A = A  If you not something twice you are back to the beginning
 

Boolean Rules 10) A + AB = A Proof: A + AB = A(1 +B) DISTRIBUTIVE LAW A + AB = A(1 +B) DISTRIBUTIVE LAW = A·1 RULE 2: (1+B)=1 = A RULE 4: A·1 = A


Boolean Rules 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

Boolean Rules 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

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