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Simplify the following expression using Boolean algebra rules:
(X + Y)’ (X’ + Y’) + (X + Y)
Answer:
Answer by student
The simplified expression is X’Y’ + XY.
Detailed answer by teachoo
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To simplify the expression, we can use the following Boolean algebra rules:
- Complement rule : X + X’ = 1 and X.X’ = 0
- Identity rule : X + 0 = X and X.1 = X
- Distributive rule : X(Y + Z) = XY + XZ and X + YZ = (X + Y)(X + Z)
- De Morgan’s rule : (X + Y)’ = X’Y’ and (XY)’ = X’ + Y’
Using these rules, we can simplify the expression as follows:
(X + Y)’ (X’ + Y’) + (X + Y)
= (X’Y’) (X’ + Y’) + (X + Y) // by De Morgan’s rule
= X’Y’X’ + X’Y’Y’ + XX + XY // by distributive rule
= X’Y’ + 0 + 0 + XY // by complement rule
= X’Y’ + XY // by identity rule
So, the simplified expression is X’Y’ + XY.
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