simplify the expression (show the steps):
x= (~A)(~B)+AB+(~A)(~B)(~C)+AB(~C)+~((A+D)(B+C+D))+~(A+B)B
To simplify the expression, we'll apply the laws of Boolean algebra and logic. Let's break down the expression step by step:
1. Simplify (~A)(~B) + AB:
- We can use the distributive law: (x)(y) + (x)(z) = (x)(y + z)
- Applying this law, (~A)(~B) + AB = (~A)(~B + B) = (~A)(1) = ~A
2. Simplify (~A)(~B)(~C) + AB(~C):
- Here, we have a common factor of AB and (~C), which we can factor out: AB(~C) = AB(1)(~C) = AB(~C)
- Simplifying further, (~A)(~B)(~C) + AB(~C) = (~A)(~B)(~C + ~C) = (~A)(~B)(~C)
3. Simplify ~((A+D)(B+C+D)):
- We can apply De Morgan's Law: ~(x + y) = ~x ~y
- Using this law, ~((A+D)(B+C+D)) = ~(A+D) ~(B+C+D) = (~A~D)(~B~C~D)
4. Simplify ~(A+B)B:
- We can again apply De Morgan's Law: ~(x + y) = ~x ~y
- Using this law, ~(A+B)B = ~(A+B)~B = (~A~B)(~B)
Now that we've simplified each part, let's combine them:
~A + (~A)(~B)(~C) + (~A~D)(~B~C~D) + (~A~B)(~B)
This is the simplified expression after applying the steps.