Tugas 5 [Dwi Nita Maulida] Aljabar Boolean

- April 05, 2023

ATURAN-ATURAN ALJABAR BOOLEAN

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

AB = BA the order of ANDing 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 have changed nothing

Aln Boolesn 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

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

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

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

END OF BOOLEAN RULES & LAWS

SUMBER : https://onlinelearning.uhamka.ac.id

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