Question
Give an alternative equivalent simple circuits for the following circuits :
✓
Answer
Let p : the switch $\mathrm{S}_1$ is closed
$q$ : the switch $S_2$ is closed
$r$ : the switch $S_3$ is closed
$\sim \mathrm{q}$ : the switch $\mathrm{S}_2{ }^{\prime}$ is closed or the switch $\mathrm{S}_2$ is open
$\sim r$ : the switch $S_3{ }^{\prime}$ is closed or the switch $S_3$ is open.
Then the symbolic form of the given circuit is :
$[p \wedge(q \vee r)] \vee(\sim r \wedge \sim q \wedge p)$.
Using the laws of logic, we have
$[p \wedge(q \vee r)] \vee(\sim r \wedge \sim q \wedge p)$
$\equiv[p \wedge(q \vee r)] \vee[\sim(r \vee q) \wedge p] \ldots .$. (By De Morgan's Law)
$\equiv[p \wedge(q \vee r)] \vee[p \wedge \sim(q \vee r)] \ldots$ (By Commutative Law)
$\equiv p \wedge[(q \vee r) \vee \sim(q \vee r)) \ldots$ (By Distributive Law)
$\equiv \mathrm{p} \wedge \mathrm{T} \ldots$... (By Complement Law)
" p ... (By Identity Law)
Hence, the alternative equivalent simple circuit is :
$q$ : the switch $S_2$ is closed
$r$ : the switch $S_3$ is closed
$\sim \mathrm{q}$ : the switch $\mathrm{S}_2{ }^{\prime}$ is closed or the switch $\mathrm{S}_2$ is open
$\sim r$ : the switch $S_3{ }^{\prime}$ is closed or the switch $S_3$ is open.
Then the symbolic form of the given circuit is :
$[p \wedge(q \vee r)] \vee(\sim r \wedge \sim q \wedge p)$.
Using the laws of logic, we have
$[p \wedge(q \vee r)] \vee(\sim r \wedge \sim q \wedge p)$
$\equiv[p \wedge(q \vee r)] \vee[\sim(r \vee q) \wedge p] \ldots .$. (By De Morgan's Law)
$\equiv[p \wedge(q \vee r)] \vee[p \wedge \sim(q \vee r)] \ldots$ (By Commutative Law)
$\equiv p \wedge[(q \vee r) \vee \sim(q \vee r)) \ldots$ (By Distributive Law)
$\equiv \mathrm{p} \wedge \mathrm{T} \ldots$... (By Complement Law)
" p ... (By Identity Law)
Hence, the alternative equivalent simple circuit is :
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