Question

Why are the dual of two eqiuivalent compound propositions also equivalent?

Answer 1

Let p and q be equivalent compound propositions involving only the operators ∧, ∨, and ¬, and T and F. Note that ¬p and ¬q are also equivalent. Use De Morgan’s laws as many times as necessary to push negations in as far as possible within these compound propositions, changing ∨ s to ∧ s, and vice versa, and changingTs to Fs, and vice versa. This shows that ¬p and ¬q are the same as p∗ and q∗except that each atomic proposition pi within them is replaced by its negation. From this, we can conclude that p∗ and q∗are equivalent because ¬p and ¬q are.

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