Mathematical reasoning practice · Medium

Consider the following statements.

Statement-I: The words "And" and "or" are connectives.

Statement-II: "There exists" and "For all" are called quantifiers.

Consider the following three statements

$p:$ Person ‘A’ passed in Mathematics exam

$q:$ Person ‘A’ passed in Physics exam

$r:$ Person ‘A’ passed in Chemistry exam

The statement $\sim \left( {\left( { \sim \left( {p \Rightarrow q} \right)} \right) \Rightarrow r} \right)$ is equivalent to

The statement $p \rightarrow(q \rightarrow p)$ is equivalent is:

The contrapositive of the statement, "I $\mathrm{x}$ is a prime number and $\mathrm{x}$ divides $\mathrm{ab}$ then $\mathrm{x}$ divides $\mathrm{a}$ or $\mathrm{x}$ ivides b", can be symbolically represented using logical connectives, on apporpriately-defined statements $p, q, r, s$, as

The negation of $x \in \mathrm{A} \cap \mathrm{B} \Rightarrow(x \in \mathrm{A}$ and $x \in \mathrm{B})$ is ......

The following statement $(\mathrm{p} \rightarrow \mathrm{q}) \rightarrow[(\sim \mathrm{p} \rightarrow \mathrm{q}) \rightarrow \mathrm{q}]$ is:

The Boolean expression $(p \wedge \sim q) \vee q \vee(\sim p \wedge q)$ is equivalent to

The contrapositive and converse of the statement ' 1 go to beach whenever it is a sunny day' is

The negation of ' $x \in \mathrm{R}$ if and only if $|x| \geq 0$ ' is ...... .

Consider the following statements :

$P$ : Suman is brilliant

$Q:$ Suman is rich

$R$ : Suman is honest

The negation of the "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

The statement $p \rightarrow(q \rightarrow p)$ is equivalent to

A relation $R$ on the set of non-zero complex numbers is defined by ${z_1}R{z_2} \Leftrightarrow \frac{{{z_1} - {z_2}}}{{{z_1} + {z_2}}}$ is real, then $R$ is

If $p, q, r, s$ and $t$ are simple statements, such that $p:$ A number is terminating.
$q:$ A number is recurring.
$r$ : A number is a rational number.
$s$ : A number is an irrational number.
$t$ : A number is a real number.
Then, the compound statement, 'A number is an irrational number if and only if it is neither non-terminating nor not recurring or rational if and only if it is terminating or if non-terminating, then must be recurring and if the number is real, it can be rational or irrational', is represented by

$\sim(p \vee q) \vee(\sim p \wedge q)$ is logical equivalent to

$(\sim p \vee q)$ is logically equal to -

The contrapositive of the statement "I go to college if its not a holiday

The contrapositive of the statement
'If 7 is greater than 5 , then 8 is greater than 6 ', is

Which of the following statement is a tautology?

${p} \vee({p} \wedge {q})$ is equivalent to

Which of the following is not a proposition?

The dual of the statement $[(p \vee q) \wedge \sim q] \vee(\sim q)$ is

The negation of $(p \wedge(\sim q)) \vee(\sim p)$ is equivalent to

The statement $B \Rightarrow((\sim A) \vee B)$ is equivalent to

The negation of the expression $q \vee((\sim q) \wedge p)$ is equivalent to

If $p, q, r$ are substatements with truth values. $T, T, F$ then the Statement $r \rightarrow(p \wedge \sim q) \vee(\sim q \wedge \sim r)$ will be

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

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

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