Discussion
Which of these correctly states De Morgan’s first theorem?
Answer: A
Correct answer: \( \overline{A + B} = \bar{A} \cdot \bar{B} \)
De Morgan’s first theorem states that the complement of a sum is the product of the complements: \[ \overline{A + B} = \bar{A} \cdot \bar{B}. \] In words, NOT (A OR B) is the same as (NOT A) AND (NOT B). The second theorem is its dual: \[ \overline{A \cdot B} = \bar{A} + \bar{B}. \] A truth table confirms the first at once. Taking A and B as 0,0 gives \( \overline{0} = 1 \) on the left and \( 1 \cdot 1 = 1 \) on the right; every other combination gives 0 on both sides. The practical rule for applying either theorem is: break the bar, change the sign. Split the overbar, and swap AND for OR or OR for AND. The theorems matter because they allow any circuit to be rebuilt using only NAND gates or only NOR gates, which is why those two are called universal.
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