Maharashtra State BoardHSC Arts 12th Board Exam
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Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit. - Mathematics and Statistics

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Sum

Simplify the following so that the new circuit has a minimum number of switches. Also, draw the simplified circuit.

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Solution

Let p: the switch S1 is closed
      q: the switch S2 is closed
      r: the switch S3 is closed
      s: the switch S4 is closed
      t: the switch S5 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
   ∼r: the switch S3′ is closed or the switch S3 is open
   ∼s: the switch S4′ is closed or the switch S4 is open
   ∼t: the switch S5′ is closed or the switch S5 is open.
Then the given circuit in symbolic form is:
[(p ∧ q) ∨ ∼r ∨ ∼s ∨ ∼t] ∧ [(p ∧ q) ∨ (r ∧ s ∧t )]
Using the laws of logic, we have, 
[(p ∧ q) ∨ ∼r ∨ ∼s ∨ ∼t] ∧ [(p ∧ q) ∨ (r ∧ s ∧t )]
≡ [(p ∧ q) ∨ ∼ (r ∧ s ∧ t)] ∧ [(p ∧ q) ∨ (r ∧ s ∧ t)] .........(By De Morgan’s Law)
≡ (p ∧ q) ∨ [∼ (r ∧ s ∧ t) ∧ (r ∧ s ∧ t)] ......(By Distributive Law)
≡ (p ∧ q) ∨ F ...........(By Complement Law)
≡ p ∧ q .......(By Identity Law)
Hence, the alternative simplified circuit is:

Concept: Application of Logic to Switching Circuits
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Balbharati Mathematics and Statistics 1 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 1 Mathematical Logic
Miscellaneous Exercise 1 | Q 13.2 | Page 34
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