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Question
Construct the new switching circuit for the following circuit with only one switch by simplifying the given circuit:
Solution
Let p : The switch S1 is closed.
q : The switch S2 is closed.
r : The switch S3 is closed.
~p: The switch 1 S′ is closed or the switch S1 is open.
~q: The switch 2 S′ is closed or the switch S2 is open.
~r : The switch 3 S′ is closed or the switch S3 is open
The logical expression corresponding to the given circuit is
[p ∧ (q ∨ r)] ∨ [~r ∧ ~q ∧ p]
≡ [p ∧ (q ∨ r)] ∨ [~(r ∨ q) ∧ p] ….(De-Morgan’s law)
≡ [p ∧ (q ∨ r)] ∨ [~(q ∨ r) ∧ p] ….(Commutative law)
≡ [p ∧ (q ∨ r)] ∨ [p ∧ ~(q ∨ r)] ….(Commutative law)
≡ p ∧ [(q ∨ r) ∨ ~(q ∨ r)] ….(Distributive law)
≡ p ∧ T ….(Complement law)
≡ p ….(Identity law)
The simplified circuit is as
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