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) ∨ (p ∧ ~q ∧ r)

Using the laws of logic, we have,

(p ∧ q ∧ ~p) ∨ (~p ∧ q ∧ r) ∨ (p ∧ q ∧ r) ∨ (p ∧ ~q ∧ r)

≡ (p ∧ ~p ∧ q) ∨ (~p ∧ q ∧ r) ∨ (p ∧ q ∧ r) y (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) = (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 :