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Maths Solve the Following Question.(3 Marks)

Mathematical Logic · Maharashtra Board STD 12 (MSBSHSE - Semi English Medium). 24 questions.

Questions

Solve the Following Question.(3 Marks)

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24 questions · self-marked practice — reveal the answer and mark yourself.

Question 13 Marks

Express the following circuits in the symbolic form. Prepare the switching table :
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Answer
Let $p :$ the switch $S_1$ is closed
$q :$ the switch $S_2$ is closed
$\sim p :$ the switch $S_1‘$ is closed or the switch $S_1$ is open
$\sim q:$ the switch $S_2‘$ is closed or the switch $S_2$ is open.
Then the symbolic form of the given circuit is :
$(p ∧ q) ∨ (\sim p) ∨ (p ∧ \sim q).$
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Question 23 Marks
Express the following circuits in the symbolic form. Prepare the switching table :
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Answer
Let $p :$ the switch $\mathrm{S}_1$ is closed
$q$ : the switch $S_2$ is closed
$\sim \mathrm{p}$ : the switch $S_1^{\prime}$ is closed or the switch $S_1$ is open
$\sim \mathrm{q}$ : the switch $\mathrm{S}_2{ }^{\prime}$ is closed or the switch $\mathrm{S}_2$ is open.
Then the symbolic form of the given circuit is :
$(p \wedge q) \vee(\sim p) \vee(p \wedge \sim q)$

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Question 63 Marks
Express the following switching circuit in the symbolic form of Logic. Construct the
switching table and interpret it.

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Question 103 Marks
Write the symbolic form of the following switching circuits construct its switching table and interpret it.
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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 circuit is:
$(p \vee \sim q) \vee(\sim p \wedge q)$
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Question 113 Marks
Give an alternative equivalent simple circuits for the following circuits :Image
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 :
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Question 123 Marks
Give an alternative equivalent simple circuits for the following circuits :
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Answer
(i) Let p : the switch $\mathrm{S}_1$ is closed
$q$ : the switch $\mathrm{S}_2$ is closed
$\sim \mathrm{p}$ : the switch $\mathrm{S}_1{ }^{\prime}$ is closed or the switch Si is open Then the symbolic form of the given circuit is :
$p \wedge(\sim p \vee q) .$
Using the laws of logic, we have,
$p \wedge(\sim p \vee q) $
$ =(p \wedge \sim p) \vee(p \wedge q) \ldots(\text { By Distributive Law }) $
$ =F \vee(p \wedge q) \ldots(\text { By Complement Law })$
$ =p \wedge q \ldots \text { (By Identity Law) }$
Hence, the alternative equivalent simple circuit is :
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Question 133 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.

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Answer
The symbolic form of the given circuit is : (p ∨ q) ∧ (q ∨ r) ∧ (r ∨ p)

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Question 143 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.

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Answer
The symbolic form of the given circuit is : [p ∨ (~p ∧ ~q)] ∨ (p ∧ q).

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Question 153 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.

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Answer
The symbolic form of the given circuit is : (p ∨ q) ∧ q ∧ (r ∨ ~p).

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Question 163 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.

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Answer
The symbolic form of the given circuit is : [p ∧ (~q ∨ r)] ∨ (~q ∧ ~ r).

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Question 173 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.

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Answer
The symbolic form of the given circuit is : (~ p ∧ q) ∨ (p ∧ ~ q).

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Question 183 Marks
Express the following circuits in the symbolic form of logic and writ the input-output table.Image
Answer

Image
Let $p$ : the switch $S_1$ is closed
$q :$ the switch $\mathrm{S}_2$ is closed
$r$ : the switch $S_3$ is closed
$\sim p$ : the switch $S_1{ }^{\prime}$ is closed or the switch $S_1$ is open
$\sim \mathrm{q}$ : 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
$I:$ the lamp $L$ is on
$(i)$ The symbolic form of the given circuit is: $p \vee(q \wedge r)=1$
$I$ is generally dropped and it can be expressed as : $p \vee(q \wedge r)$.
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Question 243 Marks
Construct the switching circuit of the following : $(\sim p \wedge q) \vee(p \wedge \sim r)$
Answer
Let p : the switch $S _1$ is closed
$q$ : the switch $S_2$ is closed
$r$ : the switch $S_3$ 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
$\sim r$ : the switch $S _3{ }^{\prime}$ is closed or the switch $S _3$ is open.
Then the switching circuits corresponding to the given statement patterns are

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