Mathematical reasoning practice · Easy

Consider the two statements:
$\left( {{S_1}} \right):\left( {p \to q} \right) \vee \left( { \sim q \to p} \right)$ is a tautology
$\left( {{S_2}} \right):\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \vee q} \right)$ is a fallacy
then,

Which of the following statement is a tautology?

Contrapositive of the statement.

If a number is divisible by 9 , then it is divisible by 3 , is

Among the statements

$(\mathrm{S} 1):(p \Rightarrow q) \vee((\sim p) \wedge q)$ is a tautology

$(\mathrm{S} 2):(q \Rightarrow p) \Rightarrow((\sim p) \wedge q)$ is a contradiction

The negation of statement $(\sim p \vee q) \wedge(\sim p \wedge \sim q)$ is -

For any three propositions $p,q$ and $r$, the proposition $\left( {p \wedge q} \right) \wedge \left( {q \wedge r} \right)$

The negation of the proposition $q \vee \sim \left( {p \wedge r} \right)$ is

If $p$ : The Earth is round, $q: 3+4=7$, then $(\sim p) \vee(\sim q)$ is

The compound statement $(P \vee Q) \wedge ( \sim P) \Rightarrow Q$ is equivalent to

If truth value of $p$ is $T, q$ is $F$, then truth values of $(p \rightarrow q)$ and $(q \rightarrow p) \vee(\sim p)$ is are respectively

Negation of the statement $p \vee(p \wedge \sim q)$ is:

Which of the following is the negation of the statement "for all $M\, > \,0$ , there exists $x \in \,S$ such that $x \ge M$ ?

The negation of $q \vee \sim \left( {p \wedge r} \right)$ is

Consider the following two propositions:
$P1:\,\, \sim (p \to \sim q)\,$
$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$
If the proposition $p \to (( \sim p) \vee q)$ is evaluated as FALSE, then:

Consider

Statement-1: $(p \wedge \sim q) \wedge(\sim p \wedge q)$ is a fallacy.

Statement-2: $(p \rightarrow q) \leftrightarrow(\sim q \rightarrow \sim p)$ is a tautology.

If $P$ and $Q$ are two statements, then which of the following compound statement is a tautology?

If $p$ and $q$ are two statements. Then negation of compound statement $(\sim p \vee q)$ is

Negation of the conditional, "If it rains, I shall go to school" is

Let $\mathrm{S}$ be a non-empty subset of $\mathrm{R}$. Consider the following statement :
$\mathrm{P}$ : There is a rational number $x \in \mathrm{S}$ such that $x>0$.
Which of the following statements is the negation of the statement P ?

Let $p$ be the proposition Mathematics is interesting and let $q$ be the proposition that Mathematics is difficult, then the symbol $p \wedge q$ means

The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is

The negation of $\sim s \vee ( \sim r \wedge s)$ is equivalent to

Consider the following statements :
P : Ramu is intelligent.
Q : Ramu is rich.
R : Ramu is not honest.
The negation of the statement “Ramu is intelligent and honest if and only if Ramu is not rich” can be expressed as :

Consider the following two statements :

P: If 7 is an odd number, then 7 is divisible by 2 .

$Q$ : If 7 is a prime number, then 7 is an odd number.

If $V_1$ is the truth value of the contrapositive of $\mathrm{P}$ and $\mathrm{V}_2$ is the truth value of contrapositive of $\mathrm{Q}$, then the ordered pair $\left(\mathrm{V}_1, \mathrm{~V}_2\right)$ equals:

Which of the following is logically equivalent to $\sim \left( { \sim p \to q} \right)$ ?

The contrapositive of $2x + 3 = 9 \Rightarrow x \ne 4$ is

Negation of the Boolean expression $p \Leftrightarrow(q \Rightarrow p)$ is

The negative of the statement $\sim \,p\, \wedge \,(p\, \vee q)\,$ is

Negation of the statement If a number is prime then it is odd' is.

Let $\mathrm{p}, \mathrm{q}, \mathrm{r}$ denote arbitrary statements. Then the logically equivalent of the statement $p \Rightarrow(q \vee r)$ is: