Question

Write the duals of each of the following.

i) $p \vee (q \wedge r)$ ii) $p \wedge (q \wedge r)$ iii) $(p \vee q) \wedge (r \vee s)$ iv) $p \wedge \sim q$ v) $(\sim p \vee q) \wedge (\sim r \wedge s)$ vi) $\sim p \wedge (\sim q \wedge (p \vee q) \wedge \sim r)$ vii) $[\sim (s \sim \vee b) \sim \wedge d] \vee [(b \wedge d) \sim]$ viii) $c \vee \{p \wedge (q \vee r)\}$ ix) $\sim p \vee (q \wedge r) \wedge d \sim (xi \{(r \wedge b) \vee d\} \wedge c$ x) $\exists \wedge (b \wedge d) (x \perp \vee (r \vee b) \wedge d \sim$

Answer 1

The dual of a Boolean expression is formed by interchanging every ∨ (OR) with ∧ (AND), and every ∧ (AND) with ∨ (OR), leaving the variables and complementation unchanged. Parts (ix) and (x) of the original question are unclear and cannot be solved.

Answer 2

To find the dual of a Boolean expression, follow these steps:

1. Identify all occurrences of the logical OR operator (∨) and the logical AND operator (∧).
2. Replace each ∨ with ∧.
3. Replace each ∧ with ∨.
4. Leave all variables (e.g., p, q, r) and complements (¬p, ~q) unchanged.

For example, the dual of $p \vee (q \wedge r)$ is $p \wedge (q \vee r)$. This process effectively swaps the roles of conjunction and disjunction in the expression.