Question

Give alternative arrangement of the switching following circuit, has minimum switches.

Answer 1

Let p: the switch S₁ is closed
      q: the switch S₂ is closed
      r: the switch S3 is closed
   ∼p: the switch S₁′ is closed or the switch S₁ is open. 
   ∼q: the switch S₂′ is closed or the switch S₂ is open.

Then the symbolic form of the given circuit is:
(p ∧ q ∧ ∼p) ∨ (∼p ∧ q ∧ r) ∨ (p ∧ q ∧ r) v (p ∧ ∼q ∧ r)

Using the laws of logic, we have,

(p ∧ q ∧ ∼p) ∨ (∼p ∧ q ∧ r) ∨ (p ∧ q ∧ r) v (p ∧ ∼q ∧ r)

≡ (p ∧ ∼p ∧ q) ∨ (∼p ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ∼ q ∧ r) ...........(By Commutative Law)

≡ (F ∧ q) ∨ (∼p  ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ∼q ∧ r) ..............(By Complement Law)
≡ F ∨ (∼p ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ∼q ∧ r) .......(By Identity Law)

≡ (∼p ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ∼q ∧ r) ..........(By Identity Law)

≡ [(∼p ∨ p) ∧ (q ∧ r)] ∨ (p ∧ ∼q ∧ r) ....(By Distributive Law)

≡ [T ∧ (q ∧ r)] ∨ (p ∧ ∼q ∧ r) ......(By Complement Law)

≡ (q ∧ r) ∨ (p ∧ ∼q ∧ r) .........(By Identity Law)

≡ [q ∨ (p ∧ ∼q)] ∧ r .........(By Distributive Law)

≡ [(q ∨ p) ∧ (q ∨ ∼q)] ∧ r ........(By Distributive Law)

≡ (q ∨ p) ∧ T] ∧ r ........(By Complement Law)

≡ (q ∨ p) ∧ r .......(By Identity Law)

≡ (p ∨ q) ∧ r ..........(By Commutative Law)

∴ the alternative arrangement of the new circuit with minimum switches is: