Mathematics · Ch 1 — Mathematical Logic

Dual

1.3.2

Dual

Two statements built only from the connectives $∨$ (or) and $∧$ (and) — possibly together with the tautology $t$ and the contradiction $c$ — are called duals of one another if one can be obtained from the other by a strict, mechanical swap: every $∨$ becomes $∧$ and every $∧$ becomes $∨$ ; every $t$ becomes $c$ and every $c$ becomes $t$ . Nothing else changes — the statement letters $p, q, r, …$ and any negations $~p, ~q, …$ stay exactly where they are; only the bare connectives and the two special symbols flip.

For example, the dual of $p ∨ q$ is $p ∧ q$ ; the dual of $t ∨ p$ is $c ∧ p$ ; and the dual of $t ∧ p$ is $c ∨ p$ .

Worked example. Write the dual of each statement.

#	Statement	Dual
i	$(p ∧ q) ∨ r$	$(p ∨ q) ∧ r$
ii	$t ∨ (p ∨ q)$	$c ∧ (p ∧ q)$
iii	$p ∧ [~q ∨ (p ∧ q) ∨ ~r]$	$p ∨ [~q ∧ (p ∨ q) ∧ ~r]$
iv	$(p ∨ q) ∧ t$	$(p ∧ q) ∨ c$
v	$(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)$	$(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)$
vi	$p ∧ q ∧ r$	$p ∨ q ∨ r$
vii	$(p ∧ t) ∨ (c ∧ ~q)$	$(p ∨ c) ∧ (t ∨ ~q)$

Misc 1Worked examples of taking a dual

Worked out. Taking a dual is a purely mechanical, symbol-by-symbol swap: the dual of p ∨ q is p ∧ q (∨ becomes ∧); the dual of t ∨ p is c ∧ p (both the connective and the constant t flip together); and the dual of t ∧ p is c ∨ p. Statement letters such as p, q, r and any negation sign ~ attached to them are left completely unchanged — only the connectives ∧/∨ and the special symbols t (tautology) and c (contradiction) are swapped, which is why duality gives a fast, purely symbolic way to generate a second valid law once a first …

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