Question

Simplify the following so that the new circuit.

Answer 1

Let p: the switch S₁ is closed
     q: the switch S₂ 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 switching circuit is:
(∼p ∨ q) ∨ (p ∨ ∼q) ∨ (p ∨ q)
Using the laws of logic, we have,
(∼p ∨ q) ∨ (p ∨ ∼q) ∨ (p ∨ q)
≡ (∼p ∨ q ∨ p ∨ q) ∨ (p ∨q)
≡ [(∼p ∨ p) ∨ (q ∨ q)] ∨ (p ∨ q) ......(By Commutative Law)
≡ (T ∨ T) ∨ (p ∨ q) ..........(By Complement Law)
≡ T ∨ (p ∨ q) .......(By Identity Law)
≡ T .......(By Identity Law)
∴ the current always flows whether the switches are open or closed. So, it is not necessary to use any switch in the circuit.
∴ the simplified form of the given circuit is: