#### 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 S_{2} is closed.

r : The switch S_{3} is closed.

~p: The switch 1 S′ is closed or the switch S_{1} is open.

~q: The switch 2 S′ is closed or the switch S_{2} is open.

~r : The switch 3 S′ is closed or the switch S_{3} 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