Tugas 5 [Muchlas 2103015057] Aljabar Bolean
Laws
& Rules of Boolean Algebra
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
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
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)
= A·1
= A
DISTRIBUTIVE LAW
RULE 2: (1+B)=1
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
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 Tugas
.png)
Komentar
Posting Komentar