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Question
Simplify the following circuit so that the new circuit has minimum number of switches. Also, draw the simplified circuit.
Solution
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 given circuit in symbolic form 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)] ...(By Associative Law)
≡ (p ∧ ∼q) ∨ [∼p ∧ (q ∨ ∼q)] ...(By Distributive Law)
≡ (p ∧ ∼q) ∨ (∼p ∧ T) ...(By Complement Law)
≡ (p ∧ ∼q) ∨ ∼p ...(By Identity Law)
≡ (p ∨ ∼p) ∧ (∼q ∨ ∼p) ...(By Distributive Law)
≡ T ∧ (∼q ∨ ∼p) ...(By Complement Law)
≡ ∼q ∨ ∼p ...(By Identity Law)
≡ ∼p ∨ ∼q ...(By Commutative Law)
Hence, the simplified circuit for the given circuit is:
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