# Give alternative arrangement of the switching following circuit, has minimum switches. - Mathematics and Statistics

Sum

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

#### Solution

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:

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 15 | Page 35