GATE CS 2025 Set 2 — Question 50
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Digital Logic → Boolean Algebra & Logic Gates → Boolean Laws & Theorems
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Question
Which of the following Boolean algebraic equation(s) is/are CORRECT?
Correct answer
(B) A B + A C + B C = A B + A C; (C) (A + C) (A + B) = A B + A C; (D) (A + B + D) (C + D) (A + C + D) (A + B + D) = A D + C D
Solution
Let's evaluate each option individually:Option (A):
LHS:
Grouping terms:
Using the absorption-like identity :
RHS:
Since , Option (A) is INCORRECT.Option (B):
This is the standard Consensus Theorem. It can be proven by expanding the term as :
Thus, Option (B) is CORRECT.Option (C):
LHS:
By applying the Consensus Theorem (as shown in Option B), .
Thus, Option (C) is CORRECT.Option (D):
LHS:
Applying De Morgan's Law ():
Grouping terms:
This matches the RHS. Thus, Option (D) is CORRECT.Final correct options are (B), (C), and (D).
LHS:
Grouping terms:
Using the absorption-like identity :
RHS:
Since , Option (A) is INCORRECT.Option (B):
This is the standard Consensus Theorem. It can be proven by expanding the term as :
Thus, Option (B) is CORRECT.Option (C):
LHS:
By applying the Consensus Theorem (as shown in Option B), .
Thus, Option (C) is CORRECT.Option (D):
LHS:
Applying De Morgan's Law ():
Grouping terms:
This matches the RHS. Thus, Option (D) is CORRECT.Final correct options are (B), (C), and (D).
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