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

Comments

Post a Comment

Popular posts from this blog

Tugas 2. Rangkuman Materi Sistem Bilangan

Image
 Tujuan Topik Bahasan  Mengulas kembali sistem bilangan desimal.  Menghitung dalam bentuk bilangan biner.  Memindahkan dari bentuk bilangan desimal ke biner dan dalam biner ke dalam desimal.  Penggunaan operasi aritmatika pada bilangan biner.  Menentukan komplemen 1 dan 2 dari sebuah bilangan biner.  Dan lain – lainnya…….. Pendahuluan Sistem Biner dan Kode – kode digital merupakan dasar untuk komputer dan elektronika digital secara umum.  Sistem bilangan biner seperti desimal, hexadesimal dan oktal juga dibahas pada bagian ini. Operasi aritmatika dengan bilangan biner akan dibahas untuk memberikan dasar pengertian bagaimana komputer dan jenis – jenis perangkat digital lain bekerja. Sistem Bilangan Desimal -> 0 ~ 9 Biner ->0 ~ 1 Oktal -> 0 ~ 7 Hexadesimal -> 0 ~ F Bilangan Desimal Dalam setiap bilangan desimal terdiri dari 10 digit, 0 sampai dengan 9 Contoh: Ungkapkan bilangan desimal 2745.214 sebagai penjumlahan nilai setiap digit. Bil...
Read more

7. Aljabar Boolean, Penyederhanaan Logika dan Peta Karnaugh

Image
Standard Forms of Boolean Expressions  Sum of Product (SOP)  Product of Sum (POS) The Sum-of-Products (SOP) Form   When two or more product terms are summed by Boolean   Conversion of a General Expression to SOP Form  Any logic expression can be change into SOP form by applying Boolean Algebra techniques The Standard SOP Form The Products-of-Sum (POS) Form  When two or more sum terms are multiplied The Standard POS Form Boolean Expression and Truth Table Converting SOP to Truth Table Examine each of the products to determine where the product is equal to a 1.  Set the remaining row outputs to 0. Converting POS to Truth Table Opposite process from the SOP expressions.  Each sum term results in a 0.  Set the remaining row outputs to 1. Converting from Truth Table to SOP and POS The Karnaugh Map The Karnaugh Map Provides a systematic method for simplifying Boolean expressions  Produces the simplest SOP or POS expression  Similar t...
Read more