De Morgan's law states which of the following?
2013
De Morgan's law states which of the following?
- A.
(X ∧ Y)' = X' ∨ Y'
- B.
(X ∨ Y)' = X' ∧ Y'
- C.
Both
- D.
None of these
Attempted by 25 students.
Show answer & explanation
Correct answer: C
Concept
De Morgan's laws describe how a NOT (complement) distributes across the two basic connectives. The governing rule is: pushing a complement inside a bracket flips the connective and negates each operand — AND becomes OR and OR becomes AND. They form a matched pair of identities, not a single one.
Application
Apply the rule to each connective in turn:
Complement of a conjunction: (X ∧ Y)' = X' ∨ Y' — the AND flips to OR and each variable is negated.
Complement of a disjunction: (X ∨ Y)' = X' ∧ Y' — the OR flips to AND and each variable is negated.
Each of the two identities just derived is individually one of these two true forms, so neither can be rejected and neither alone is the full law.
Cross-check
Verify by truth values. Take X = 1, Y = 0:
(X ∧ Y)' = (1 ∧ 0)' = 0' = 1, and X' ∨ Y' = 0 ∨ 1 = 1 — they match.
(X ∨ Y)' = (1 ∨ 0)' = 1' = 0, and X' ∧ Y' = 0 ∧ 1 = 0 — they match.
Both identities hold for every assignment, so the complete statement of De Morgan's law is the pair together.