simplify the following equation using boolean laws and rules:
x=~(A(+)B) + ~(AB)+~((A(+)B)AC)+(~A)C
(+) means plus in the circle
~ means not
Do you get
~A + ~B + ~C
?
i got two things to say:
1) how did you get ~A + ~B + ~C
2) my friend of mines got ~(ABC) for this exact question, is my friend correct?
The two are equivalent.
how did the other Anonymous get ~A + ~B + ~C
To simplify the given equation using Boolean laws and rules, we will apply various logic operations and simplifications step by step. Let's break down the given equation:
x = ~(A(+)B) + ~(AB) + ~((A(+)B)AC) + (~A)C
Step 1: Apply the De Morgan's Laws
De Morgan's Laws state:
- ~(A + B) = ~A * ~B (1)
- ~(A * B) = ~A + ~B (2)
Using equation (1), simplify the first term ~(A(+)B):
~(A(+)B) = ~A * ~B
x = ~A * ~B + ~(AB) + ~((A(+)B)AC) + (~A)C
Step 2: Apply De Morgan's Laws again
Using equation (2), simplify the second term ~(AB):
~(AB) = ~A + ~B
x = ~A * ~B + (~A + ~B) + ~((A(+)B)AC) + (~A)C
Step 3: Distribute the negation ~(A(+)B)AC
~((A(+)B)AC) = ~(A(+)B) * ~A * ~C
x = ~A * ~B + (~A + ~B) + ~(A(+)B) * ~A * ~C + (~A)C
Step 4: Distribute the negation ~(A(+)B) * ~A * ~C
~(A(+)B) * ~A * ~C = (~A * ~B) * ~A * ~C
x = ~A * ~B + (~A + ~B) + (~A * ~B) * ~A * ~C + (~A)C
Step 5: Apply the idempotent law
In Boolean algebra, the idempotent law states that:
- X + X = X
- X * X = X
Apply the idempotent law to simplify the equation further:
x = ~A * ~B + ~A + ~B + (~A * ~B) * ~A * ~C + (~A)C
Step 6: Apply the distributive law
The distributive law states that:
- X * (Y + Z) = X * Y + X * Z
- X + (Y * Z) = (X + Y) * (X + Z)
Apply the distributive law to simplify the equation:
x = ~A * (~B + 1) + ~B + (~A * ~B) * ~A * ~C + (~A)C
Since ~B + 1 = 1 (complement law), and ~A * ~B * ~A = 0 (one of the variables is complemented), we can further simplify:
x = ~A + ~B + 0 * ~C + (~A)C
Step 7: Simplify the zero and identity operations
- 0 * X = 0
- X + 0 = X
Applying these simplifications:
x = ~A + ~B + 0 + (~A)C
Step 8: Simplify the equation further
- 0 + X = X
Applying this simplification:
x = ~A + ~B + (~A)C
Therefore, the simplified equation is x = ~A + ~B + (~A)C.