Answer
DEFINITIONS
De Morgan’s law:
$∼(p∧q)≡∼p∨∼q$
$∼(p∨q)≡∼p∧∼q$
Double negative law:
$∼(∼p)≡p$
Idempotent laws:
$p∨p≡p$
$p∧p≡p$
Work Step by Step
(a) By the definition of the Sheffer Stroker:
$P∣Q≡∼(P∧Q)$
$(P∣Q)∣(P∣Q)
≡∼((P∣Q)∧(P∣Q))$
$≡∼(∼(P∧Q)∧∼(P∧Q))$
$≡∼(∼(P∧Q))∨∼(∼(P∧Q))$
$≡(P∧Q)∨∼(∼(P∧Q))$
$≡(P∧Q)∨(P∧Q)$
$≡P∧Q$
We have derived that (P∣Q)∣(P∣Q) is a logically equivalent with
P∧Q.
$P∧Q≡(P∣Q)∣(P∣Q)$
(b):
$P∧(∼Q∨R)≡P∧(∼Q∨∼(∼R))$
$≡P∧∼(Q∧(∼R))$
$≡P∧∼(Q∧(∼(R∧R)))$
$≡P∧∼(Q∧(R∣R))$
$≡P∧(Q∣(R∣R))$
We have derived that $(P∣(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$ is a logically equivalent with $P∧(∼Q∨R)$, while
$(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$ contains only Sheffer strokes.
RESULT:
(a)
$P∧Q≡(P∣Q)∣(P∣Q)$
(b)
$(P∣(Q∣(R∣R)))∣(P∣(Q∣(R∣R)))$
