Question
Simplify the following so that the new circuit circuit.
✓
Answer
Let p : the switch $S _1$ is closed
q : the switch $S _2$ is closed
$\sim p$ : the switch $S _1{ }^{\prime}$ is closed or the switch $S _1$ is open
$\sim q$ : the switch $S _2{ }^{\prime}$ is closed or the switch $S _2$ is open.
Then the symbolic form of the given switching circuit is :
$(\sim p \vee q) \vee(p \vee \sim q) \vee(p \vee q)$
Using the laws of logic, we have,
$(\sim p \vee q) \vee(p \vee \sim q) \vee(p \vee q)$
$\equiv(\sim p \vee q \vee p \vee \sim q) \vee(p \vee q)$
$\equiv[(\sim p \vee p) \vee(q \vee \sim q)] \vee(p \vee q) \ldots \text { (By Commutative Law) }$
$\equiv(T \vee T) \vee(p \vee q) \ldots \text { (By Complement Law) }$
$\equiv T \vee(p \vee q) \ldots \text { (By Identity Law) }$
$\equiv T \ldots \text { (By Identity Law) }$
$\therefore$ the current always flows whether the switches are open or closed. So, it is not necessary to use any switch in the circuit.
$\therefore$ the simplified form of given circuit is:
q : the switch $S _2$ is closed
$\sim p$ : the switch $S _1{ }^{\prime}$ is closed or the switch $S _1$ is open
$\sim q$ : the switch $S _2{ }^{\prime}$ is closed or the switch $S _2$ is open.
Then the symbolic form of the given switching circuit is :
$(\sim p \vee q) \vee(p \vee \sim q) \vee(p \vee q)$
Using the laws of logic, we have,
$(\sim p \vee q) \vee(p \vee \sim q) \vee(p \vee q)$
$\equiv(\sim p \vee q \vee p \vee \sim q) \vee(p \vee q)$
$\equiv[(\sim p \vee p) \vee(q \vee \sim q)] \vee(p \vee q) \ldots \text { (By Commutative Law) }$
$\equiv(T \vee T) \vee(p \vee q) \ldots \text { (By Complement Law) }$
$\equiv T \vee(p \vee q) \ldots \text { (By Identity Law) }$
$\equiv T \ldots \text { (By Identity Law) }$
$\therefore$ the current always flows whether the switches are open or closed. So, it is not necessary to use any switch in the circuit.
$\therefore$ the simplified form of given circuit is:
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