Boolean Algebra - ISC Class 12 Computer Science Questions with Answers, Page 10

Principle of Duality: if a Boolean equation or theorem is true, then its dual is also true. The dual is obtained by interchanging every OR (+) with AND (.) and every 0 with 1, while the variables (and their complements) are left unchanged. Example: $A + 0 = A$ has the dual $A \cdot 1 = A$.

$F(A,B,C) = (B + C') \cdot (A' + B)$. Add the missing variable to each term using $X \cdot X' = 0$: $(B + C') = (B + C' + A \cdot A') = (A + B + C') \cdot (A' + B + C')$ $(A' + B) = (A' + B + C \cdot C') = (A' + B + C) \cdot (A' + B + C')$ Combining and removing the repeated term $(A' + B + C')$: $F(A,B,C) = (A + B + C') \cdot (A' + B + C) \cdot (A' + B + C')$ $= \pi(1, 4, 5)$ (verified with the boolean tool).

To prove: $(x' + z) + [(y' + z) \cdot (x' + y)]' = 1$. Verified with the boolean tool: the expression is a tautology (always 1).

Minterm: a product (AND) term in which every variable of the function appears exactly once, in complemented or uncomplemented form; it is 1 for exactly one combination of inputs. Maxterm: a sum (OR) term in which every variable appears exactly once, in complemented or uncomplemented form; it is 0 for exactly one combination of inputs. For $P = 0, Q = 1, R = 1, S = 0$ (binary 0110 = 6): Minterm ($m_6$): $P'QRS'$ (a variable with value 0 is complemented, value 1 is uncomplemented) Maxterm ($M_6$): $P + Q' + R' + S$ (a variable with value 1 is complemented, value 0 is uncomplemented)

Logic gate diagram for $F = (P + R)(P + S)(Q + S)$: three 2-input OR gates with inputs (P, R), (P, S) and (Q, S) feed one 3-input AND gate whose output is F.

$F' = a'bc'$ $F = (b' + c) + a$ $F' = [(b' + c) + a]' = (b' + c)' \cdot a'$ (De Morgan's law) $= (b'' \cdot c') \cdot a' = b \cdot c' \cdot a'$ (De Morgan's law, involution law) So $F' = a'bc'$ (verified with the boolean tool).

Converse: interchange the antecedent and the consequent of $\sim P \Rightarrow Q$. Converse: $Q \Rightarrow \sim P$.