By the definitions of equivalence and implication,
\[\begin{align*}
& \lnot (P \land Q) \iff \lnot P \lor \lnot Q
\\[1ex]
& \quad \equiv ((\lnot (P \land Q) ) \implies (\lnot P \lor \lnot Q))
\land ((\lnot P \lor \lnot Q ) \implies (\lnot (P \land Q)))
\\[1ex]
& \quad \equiv (\lnot ( \lnot (P \land Q)) \lor (\lnot P \lor \lnot Q))
\land (\lnot (\lnot P \lor \lnot Q) \lor (\lnot (P \land Q)))
\\[2ex]
& \lnot (P \lor Q) \iff \lnot P \land \lnot Q
\\[1ex]
& \quad \equiv ((\lnot (P \lor Q)) \implies (\lnot P \land \lnot Q))
\land ((\lnot P \land \lnot Q) \implies (\lnot (P \lor Q)))
\\[1ex]
& \quad \equiv (\lnot (\lnot (P \lor Q)) \lor (\lnot P \land \lnot Q))
\land (\lnot (\lnot P \land \lnot Q) \lor (\lnot (P \lor Q))),
\end{align*}
\]
and those are tautologies, per
(define (boolean-product length)
"Return a list of all Boolean sequences of LENGTH."
(let more-rows ((row (- (expt 2 length) 1)))
(if (< row 0) (list)
(cons (let more-columns ((column (- length 1)))
(if (< column 0) (list)
(cons (even? (quotient row (expt 2 column)))
(more-columns (- column 1)))))
(more-rows (- row 1))))))
(define (truth-table procedure arity)
"Return a truth table for PROCEDURE with ARITY."
(map (lambda (inputs)
(append inputs (list (apply procedure inputs))))
(boolean-product arity)))
(define (tautology? procedure arity)
(not (member #f (map (lambda (row)
(car (reverse row)))
(truth-table procedure arity)))))
(and (tautology? (lambda (p q)
(and (or (not (not (and p q)))
(or (not p) (not q)))
(or (not (or (not p) (not q)))
(not (and p q)))))
2)
(tautology? (lambda (p q)
(and (or (not (not (or p q)))
(and (not p) (not q)))
(or (not (and (not p) (not q)))
(not (or p q)))))
2))
#t
so De Morgan’s laws are always ‘true’. \(\blacksquare\)