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Mathematical Logic · Maharashtra Board STD 12 (MSBSHSE - Semi English Medium). 17 questions.

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Question 11 Mark
Construct the truth table for each of the following statement patterns.
i) $\quad p \rightarrow(q \rightarrow p)$
ii) $(\sim p \vee q) \leftrightarrow \sim(p \wedge q)$
iii) $\sim(\sim p \wedge \sim q) \vee q$
iv) $[(p \wedge q) \vee r ] \wedge[\sim r \vee(p \wedge q)]$
v) $[(\sim p \vee q) \wedge(q \rightarrow r )] \rightarrow(p \rightarrow r )$
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Question 21 Mark
Write the negations of the following.
i) Price increases
ii) 0! ≠ 1
iii) 5 + 4 = 9
Answer
i) Price does not increase
ii) 0! = 1
iii) 5 + 4 ≠ 9
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Question 31 Mark
If statements p, q are true and r, s are false, determine the truth values of the following. i) i) $\sim p \wedge(q \vee \sim r )$
ii) $(p \wedge \sim r ) \wedge(\sim q \vee s )$
iii) $\sim(p \rightarrow q) \leftrightarrow( r \wedge s )$
iv) $(\sim p \rightarrow q) \wedge( r \leftrightarrow s )$
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Question 41 Mark
Write the truth values of the following statements.
i) 3 is a prime number and 4 is a rational number.
ii) All flowers are red or all cows are black.
iii) If Mumbai is in Maharashtra then Delhi is the capital of India.
iv) Milk is white if and only if the Sun rises in the West.
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Question 51 Mark
Express the following compound statements symbolically without examining the truth values.
i) 2 is an even number and 25 is a perfect square.
ii) A school is open or there is a holiday.
iii) Delhi is in India but Dhaka is not in Srilanks.
iv) 3 + 8 ≥ 12 if and only if 5 × 4 ≤ 25.
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Question 61 Mark
Construct switching circuits of the following.
i) $\quad[(p \vee(\sim p \wedge q)] \vee[(\sim q \wedge r ) \vee \sim p]$
ii) $(p \wedge q \wedge r ) \vee[p \vee(q \wedge \sim r)]$
iii) $\quad[(p \wedge r) \vee(\sim q \wedge \sim r)] \vee(\sim p \wedge \sim r)$
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Question 71 Mark
Which of the following sentences are statements in logic ? Write down the truth values of
the statements.
i) $6 × 4 = 25$
ii) $x + 6 = 9$
iii) What are you doing ?
iv) The quadratic equation $x^2 – 5x + 6 = 0$ has $2$ real roots.
v) Please, sit down
vi) The Moon revolves around the earth.
vii) Every real number is a complex number.
viii) He is honest.
ix) The square of a prime number is a prime number.
Answer
i) It is a statement which is false, hence its truth value is F.
ii) It is an open sentence hence it is not a statement.
iii) It is an interrogative hence it is not a statement.
iv) It is a statement which is true hence its truth value is T.
v) It is a request hence it is not a statement.
vi) It is a statement which is true, hence its truth value is T.
vii) It is a statement which is true, hence its truth value is T.
viii) It is open sentence, hence it is not a statement.
ix) It is a statement which is false, hence its truth value is F
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Question 81 Mark
Write the converse, inverse and contrapositive of the following statements.
i) If a function is differentiable then it is continuous.
ii) If it rains then the match will be cancelled
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Question 91 Mark
Write the negations of the following statements.
i) All natural numbers are rational.
ii) Some students of class $X$ are sixteen year old.
iii) $\exists n \in N$ such that $n+8>11$
iv) $\forall x \in N , 2 x+1$ is odd
Answer
i) Some natural numbers are not rationals.
ii) No student of class $X$ is sixteen year old.
iii) $\forall n \in N , n+8 \leq 11$
iv) $\exists x \in N$ such hat $2 x+1$ is not odd
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Question 101 Mark
Write the negations of the following.
i) 3 + 3 < 5 or 5 + 5 = 9
ii) 7 > 3 and 4 > 11
iii) The number is neither odd nor perfect square.
iv) The number is an even number if and only if it is divisible by 2.
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Question 111 Mark
Write the duals of each of the following:
i) $(p \wedge q) \vee r$
ii) $t \vee(p \vee q)$
iii) $p \wedge[\sim q \vee(p \wedge q) \vee \sim r]$
iv) $(p \vee q) \wedge t$
v) $(p \vee q) \vee r=p \vee(q \vee r)$
vi) $p \wedge q \wedge r$
vii) $(p \wedge t) \vee(c \wedge \sim q)$
Answer
i) $\quad(p \vee q) \wedge r$
ii) $\quad c \wedge(p \wedge q)$
iii) $p \vee[(\sim q \wedge(p \vee q) \wedge \sim r]$
iv) $(p \wedge q) \vee c$
v) $(p \wedge q) \wedge r=p \wedge(q \wedge r)$
vi) $p \vee q \vee r$
vii) $(p \vee c) \wedge( t \vee \sim q)$
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Question 121 Mark
If $A=\{1,2,3,4,5,6,7\}$, determine the truth value of the following.
i) $\exists x \in A$ such that $x-4=3$
ii) $\forall x \in A , \quad x+1 \geq 3$
iii) $\forall x \in A , 8-x \leq 7$
iv) $\exists x \in A$, such that $x+8=16$
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Question 131 Mark
Using truth tables, prove the following logical equivalences
i) $(p \wedge q) \equiv \sim(p \rightarrow \sim q)$
ii) $(p \leftrightarrow q) \equiv(p \wedge q) \vee(\sim p \wedge \sim q)$
iii) $(p \wedge q) \rightarrow r \equiv p \rightarrow(q \rightarrow r )$
iv) $p \rightarrow(q \vee r) \equiv(p \rightarrow$ q $) \vee(p \rightarrow r )$
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Question 141 Mark
Construct the truth table for each of the following: $(iv) [(p ∧ q) ∨ r] ∧ [~r ∨ (p ∧ q)]$
Answer
$p$ $q$ $r$ $p ∧ q$ $(p ∧ q) ∨ q$ $~r$ $\sim r \vee(p \wedge q)$ $[(p \wedge q) \vee r) \wedge(\sim r \vee(p \wedge q)]$
$T$ $T$ $T$ $T$ $T$ $F$ $T$ $T$
$T$ $T$ $F$ $T$ $T$ $T$ $T$ $T$
$T$ $F$ $T$ $F$ $T$ $F$ $F$ $F$
$T$ $F$ $F$ $F$ $F$ $T$ $T$ $F$
$F$ $T$ $T$ $F$ $T$ $F$ $F$ $F$
$F$ $T$ $F$ $F$ $F$ $T$ $T$ $F$
$F$ $F$ $T$ $F$ $T$ $F$ $F$ $F$
$F$ $F$ $F$ $F$ $F$ $T$ $T$ $F$
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Question 151 Mark
Construct the truth table for each of the following:
(iii) ~(~p ∧ ~q) ∨ q
Answer
pq~p~q~ p ∧ ~ q ~ (~ p ∧ ~ q)~ (~ p ∧ ~ q) ∧ q
TTFFFTT
TFFTFTT
FTTFFTT
FFTTTTT
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Question 161 Mark
Construct the truth table for each of the following statement patterns.
i) $\quad p \rightarrow(q \rightarrow p)$
Answer
pqq $\rightarrow$ pp $\rightarrow$(q $\rightarrow$ p)
TTTT
TFTT
FTFT
FFTT
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Question 171 Mark
Construct the truth table for each of the following statement patterns.
(ii) (~p ∨ ~q) ↔ [~(p ∧ q)]
Answer
pq~p~qp ∧ q ~ (p ∧ q)~p ∨ ~ q$\begin{array}{c}(\sim p \vee \sim q) \leftrightarrow {(\sim(p \wedge q))}\end{array}$
TTFFFTFT
TFFTTFTT
FTTFTFTT
FFTTTFTT
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