Tugas 5 [Muhammad Iqbal Burhanuddin] Aljabar Boolean

 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 

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

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