1.
If $$S\left( {p,q,r} \right) = \left( { \sim p} \right) \vee \left[ { \sim \left( {q \wedge r} \right)} \right]$$ is a compound statement, then $$S\left( { \sim p, \sim q, \sim r} \right)$$ is A.
$$ \sim S\left( {p,q,r} \right)$$
B.
$$S\left( {p,q,r} \right)$$
C.
$$p \vee \left( {q \wedge r} \right)$$
D.
$$p \vee \left( {q \vee r} \right)$$
Answer :
$$p \vee \left( {q \vee r} \right)$$
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$$\eqalign{
& S\left( {p,q,r} \right) = \left( { \sim p} \right) \vee \left[ { \sim \left( {q \wedge r} \right)} \right] \cr
& S\left( { \sim p, \sim q, \sim r} \right) = \sim \left( { \sim p} \right) \vee \left[ { \sim \left( { \sim q \, \wedge \sim r} \right)} \right] \cr
& = p \vee \left[ { \sim \left( { \sim q} \right) \vee \sim \left( { \sim r} \right)} \right] = p \vee \left( {q \vee r} \right) \cr} $$
2.
Let $$p$$ be the statement “$$x$$ is an irrational number”, $$q$$ be the statement “$$y$$ is a transcendental number”, and $$r$$ be the statement “$$x$$ is a rational number if $$f y$$ is a transcendental number”.Statement - 1 : $$r$$ is equivalent to either $$q$$ or $$p$$Statement - 2 : $$r$$ is equivalent to $$ \sim \left( {p \leftrightarrow \sim q} \right).$$ A.
Statement - 1 is false, Statement - 2 is true
B.
Statement - 1 is true, Statement - 2 is true ; Statement - 2 is
a correct explanation for Statement - 1
C.
Statement - 1 is true, Statement - 2 is true ; Statement - 2
is not a correct explanation for Statement - 1
D.
none of these
Answer :
none of these
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$$p$$ : $$x$$ is an irrational number
$$q$$ : $$y$$ is a transcendental number
$$r$$ : $$x$$ is a rational number if $$f y$$ is a transcendental number.
clearly $$r: \sim p \leftrightarrow q$$
Let us use truth table to check the equivalence of $$‘r’$$ and $$‘q$$ or $$p’,'r'$$ and $$ \sim \left( {p \leftrightarrow \sim q} \right)$$
1
2
3
$$p$$
$$q$$
$$ \sim p$$
$$ \sim q$$
$$ \sim p \leftrightarrow q$$
$$q{\text{ or }}p$$
$$p \leftrightarrow \, \sim q$$
$$ \sim \left( {p \leftrightarrow \, \sim q} \right)$$
T
T
F
F
F
T
F
T
T
F
F
T
T
T
T
F
F
T
T
F
T
T
T
F
F
F
T
T
F
F
F
T
From columns (1), (2) and (3), we observe, none of the these statements are equivalent to each other.
∴ Statement las well as statement 2 both are false.
∴ None of the options is correct.
3.
ConsiderStatement - 1 : $${\left( {{p^ \wedge } \sim q} \right)^ \wedge }\left( { \sim {p^ \wedge }q} \right)$$ is a fallacy.Statement - 2 : $$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)$$ is a tautology. A.
Statement - 1 is true; Statement - 2 is true;
Statement - 2 is a correct explanation for Statement - 1.
B.
Statement - 1 is true; Statement - 2 is true; Statement - 2 is
not a correct explanation for Statement - 1.
C.
Statement - 1 is true; Statement - 2 is false.
D.
Statement - 1 is false; Statement - 2 is true.
Answer :
Statement - 1 is true; Statement - 2 is true; Statement - 2 is
not a correct explanation for Statement - 1.
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Statement - 2 : $$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right) \equiv \left( {p \to q} \right) \leftrightarrow \left( {p \to q} \right)$$Statement - 1 : $${\left( {{p^ \wedge } \sim q} \right)^ \wedge }\left( { \sim {p^ \wedge }q} \right)$$
4.
Which of the following is not logically equivalent to the proposition : “A real number is either rational or irrational.” A.
If a number is neither rational nor irrational then it is not real
B.
If a number is not a rational or not an irrational, then it is not real
C.
If a number is not real, then it is neither rational nor irrational
D.
If a number is real, then it is rational or irrational
Answer :
If a number is not a rational or not an irrational, then it is not real
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It is correct.
5.
The negation of $$ \sim s \vee \left( { \sim r \wedge s} \right)$$ is equivalent to: A.
$$s \vee \left( {r \vee \sim s} \right)$$
B.
$${s \wedge r}$$
C.
$${s \wedge \sim r}$$
D.
$$s \wedge \left( {r \wedge \sim s} \right)$$
Answer :
$${s \wedge r}$$
View Solution
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$$\eqalign{
& \sim \left[ { \sim s \vee \left( { \sim r \wedge s} \right)} \right] \cr
& = s \wedge \sim \left( { \sim r \wedge s} \right) \cr
& = s \wedge \left( {r \vee \sim s} \right) \cr
& = \left( {s \wedge r} \right) \vee \left( {s \wedge \sim s} \right) \cr
& = \left( {s \wedge r} \right) \vee 0 \cr
& = s \wedge r \cr} $$
6.
The contrapositive of $$\left( {p \vee q} \right) \Rightarrow r{\text{ is}}$$ A.
$$r \Rightarrow \left( {p \vee q} \right)$$
B.
$$ \sim r \Rightarrow \left( {p \vee q} \right)$$
C.
$$ \sim r \Rightarrow \sim p \wedge \sim q$$
D.
$$p \Rightarrow \left( {q \vee r} \right)$$
Answer :
$$ \sim r \Rightarrow \sim p \wedge \sim q$$
View Solution
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Contrapositive of $$p \Rightarrow q{\text{ is }} \sim q \Rightarrow \, \sim p$$
7.
Let $$p, q$$ and $$r$$ be any three logical statements. Which of the following is true ? A.
$$ \sim \left[ {p \wedge \left( { \sim q} \right)} \right] \equiv \left( { \sim p} \right) \wedge q$$
B.
$$ \sim \left[ {\left( {p \vee q} \right) \wedge \left( { \sim r} \right)} \right. \equiv \left( { \sim p} \right) \vee \left( { \sim q} \right) \vee \left( { \sim r} \right)$$
C.
$$ \sim \left[ {p \vee \left( { \sim q} \right)} \right] \equiv \left( { \sim p} \right) \wedge q$$
D.
$$ \sim \left[ {p \vee \left( { \sim q} \right)} \right] \equiv \left( { \sim p} \right) \wedge {\sim q}$$
Answer :
$$ \sim \left[ {p \vee \left( { \sim q} \right)} \right] \equiv \left( { \sim p} \right) \wedge q$$
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Statement given in option $$\left( C \right)$$ is correct.
8.
Which of the following is a contradiction ? A.
$$\left( {p \wedge q} \right) \wedge \sim \left( {p \vee q} \right)$$
B.
$$p \vee \left( { - p \wedge q} \right)$$
C.
$$\left( {p \Rightarrow q} \right) \Rightarrow p$$
D.
None of these
Answer :
$$\left( {p \wedge q} \right) \wedge \sim \left( {p \vee q} \right)$$
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$$p$$
$$q$$
$$p \wedge q$$
$$p \vee q$$
$$ \sim \left( {p \vee q} \right)$$
$$\left( {p \wedge q} \right) \wedge \sim \left( {p \vee q} \right)$$
T
T
T
T
F
F
T
F
F
T
F
F
F
T
F
T
F
F
F
F
F
F
T
F
$$\therefore \left( {p \wedge q} \right) \wedge \left( { \sim \left( {p \vee q} \right)} \right)$$ is a contradiction.
9.
Consider the two statements $$P :$$ He is intelligent and $$Q :$$ He is strong. Then the symbolic form of the statement ‘‘It is not true that he is either intelligent or strong’’ is A.
$$ \sim P \vee Q$$
B.
$$ \sim P \wedge \sim Q$$
C.
$$ \sim P \wedge Q$$
D.
$$ \sim \left( {P \vee Q} \right)$$
Answer :
$$ \sim \left( {P \vee Q} \right)$$
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Given : $$P :$$ He is intelligent.
10.
The negation of $$\left( {p \vee \sim q} \right) \wedge q{\text{ is}}$$ A.
$$\left( { \sim p \vee q} \right) \wedge \sim q$$
B.
$$\left( {p \wedge \sim q} \right) \vee q$$
C.
$$\left( { \sim p \wedge q} \right) \vee \sim q$$
D.
$$\left( {p \wedge \sim q} \right) \vee \sim q$$
Answer :
$$\left( { \sim p \wedge q} \right) \vee \sim q$$
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$$ \sim \left\{ {\left( {p \vee \left( { \sim q} \right)} \right) \wedge q} \right\} = \left( { \sim \left( {p \vee \left( { \sim q} \right)} \right)} \right) \vee \left( { \sim q} \right)$$