Tugas 4 Rangkuman Materi Aljabar Boolean
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
Boolean Rules
1) A + 0 = A
- In math if you add 0 you have changed nothing
- In Boolean Algebra ORing with 0 changes nothing
- ORing with 1 must give a 1 since if any input is 1 an OR gate will give a 1
- In math if 0 is multiplied with anything you get 0. If you AND anything with 0 you get 0
- ANDing anything with 1 will yield the anything
- ORing with itself will give the same result
- Either A or A must be 1 so A + A =1
- ANDing with itself will give the same result
- In digital Logic 1 =0 and 0 =1, so AA=0 since one of the inputs must be 0.
- If you not something twice you are back to the beginning
Proof:
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
11)A + AB = A + B
- If A is 1 the output is 1 , If A is 0 the output is B
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
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