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Simplify the following expression.
Y = AB + A(B + C) + B(B + C)
Y = AB + A(B + C) + B(B + C)
- a)Y = A + B + C
- b)Y = B + AC
- c)Y = A + BC
- d)Y = AB + C
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Simplify the following expression.Y = AB + A(B + C) + B(B + C)a)Y = A ...
Expression Simplification
To simplify the expression Y = AB + A(B + C) + B(B + C), we will use Boolean algebra principles.
Step 1: Expand the Expression
- Start with Y = AB + A(B + C) + B(B + C).
- Expand A(B + C) and B(B + C):
- A(B + C) = AB + AC
- B(B + C) = BB + BC = B + BC (since BB = B)
Combining these gives:
Y = AB + (AB + AC) + (B + BC)
Step 2: Combine Like Terms
- Combine the terms:
- Y = AB + AB + AC + B + BC
- Simplifying further, we get Y = AB + B + AC + BC
Step 3: Apply Absorption Law
- According to the absorption law, AB + B = B.
- Therefore, we can simplify Y to:
- Y = B + AC + BC
Step 4: Factor Out Common Terms
- Notice that in B + AC + BC, we can factor out B:
- Y = B(1 + C) + AC
- Since 1 + C = 1 in Boolean algebra, we simplify to:
- Y = B + AC
Final Result
The simplified expression is Y = B + AC, which corresponds to option 'B'.
Conclusion
- The expression simplifies to B + AC, confirming that option 'B' is indeed the correct answer.
To simplify the expression Y = AB + A(B + C) + B(B + C), we will use Boolean algebra principles.
Step 1: Expand the Expression
- Start with Y = AB + A(B + C) + B(B + C).
- Expand A(B + C) and B(B + C):
- A(B + C) = AB + AC
- B(B + C) = BB + BC = B + BC (since BB = B)
Combining these gives:
Y = AB + (AB + AC) + (B + BC)
Step 2: Combine Like Terms
- Combine the terms:
- Y = AB + AB + AC + B + BC
- Simplifying further, we get Y = AB + B + AC + BC
Step 3: Apply Absorption Law
- According to the absorption law, AB + B = B.
- Therefore, we can simplify Y to:
- Y = B + AC + BC
Step 4: Factor Out Common Terms
- Notice that in B + AC + BC, we can factor out B:
- Y = B(1 + C) + AC
- Since 1 + C = 1 in Boolean algebra, we simplify to:
- Y = B + AC
Final Result
The simplified expression is Y = B + AC, which corresponds to option 'B'.
Conclusion
- The expression simplifies to B + AC, confirming that option 'B' is indeed the correct answer.
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Community Answer
Simplify the following expression.Y = AB + A(B + C) + B(B + C)a)Y = A ...
Analysis:
Y = AB + A(B + C) + B(B + C)
= AB + AB + AC + B + BC
Since AB + AB = AB, we get:
Y = AB + AC + B (1 + C)
Since 1 + X (any variable) = X, we get:
Y = AB + AC + B
Y = B(1 + A) + AC
Y = B + AC
Y = AB + A(B + C) + B(B + C)
= AB + AB + AC + B + BC
Since AB + AB = AB, we get:
Y = AB + AC + B (1 + C)
Since 1 + X (any variable) = X, we get:
Y = AB + AC + B
Y = B(1 + A) + AC
Y = B + AC
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