# Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit. - Mathematics and Statistics

Sum

Simplify the following so that the new circuit has a 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:

Concept: Application of Logic to Switching Circuits
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#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 1 Mathematical Logic
Miscellaneous Exercise 1 | Q 13.1 | Page 34