Simplify the following Boolean expression using Boolean algebra rules:
(A + B) . (A' + B') + A
Answer:
Answer by student
The simplified expression is A .
The steps are:
- (A + B) . (A’ + B’) + A
- AA’ + AB’ + BA’ + BB’ + A (Distributive law)
- 0 + AB’ + BA’ + 0 + A (Complement law)
- AB’ + BA’ + A (Identity law)
- A(B’ + A’) + A (Distributive law)
- A(0) + A (Complement law)
- 0 + A (Identity law)
- A
Detailed answer by teachoo
To simplify the expression, we need to use some Boolean algebra rules. Here are some of the rules we will use:
- Complement law : X + X’ = 1 and X.X’ = 0
- Identity law : X + 0 = X and X.1 = X
- Distributive law : X(Y + Z) = XY + XZ and X + YZ = (X + Y)(X + Z)
Let’s see how to apply these rules step by step:
- The given expression is (A + B) . (A’ + B’) + A .
- We can use the distributive law to expand the product of the first two terms as follows: AA’ + AB’ + BA’ + BB’ + A .
- We can use the complement law to simplify the terms AA’ and BB’ as follows: 0 + AB’ + BA’ + 0 + A .
- We can use the identity law to simplify the terms 0 as follows: AB’ + BA’ + A .
- We can use the distributive law again to factor out A from the first two terms as follows: A(B’ + A’) + A .
- We can use the complement law again to simplify the term (B’ + A’) as follows: A(0) + A .
- We can use the identity law again to simplify the term A(0) as follows: 0 + A .
- We can use the identity law one more time to simplify the term 0 as follows: A .
This is the final simplified expression. So, the correct answer is A.
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