Validity of a Statement

Contrapositive of statement pattern $\left(~p\vee q\right)\to \left(p\wedge ~q\right)$ is

Which of the following Boolean expression is a tautology?

Let $p$ and $q$ be any two propositions.
Statement I: $(p\to q)\leftrightarrow q\vee ~p$ is a tautology.

Statement II: $~(~p\wedge q)\wedge (p\vee q)\leftrightarrow p$ is a fallacy.

The statement $[p\wedge (p\to q\left)\right]\to q$, is

The negation of $A\to \left(A\vee ~B\right)$ is

Which of the following statement is a tautology?

Let $p$ and $q$ be two statements, then $\left(p\vee q\right)\vee ~p$ is?

Which of the following statement is a tautology

The contrapositive of $\left(p\vee q\right)\Rightarrow r$ is

The statement $\left(p\to q\right)\to \left[\left(~p\to q\right)\to q\right]$ is

Which one of the following is a tautology?

Prove by the method of contradiction that each of the numbers given below is an irrational number.

$\sqrt{39}$

Prove by the method of contradiction that each of the numbers given below is an irrational number.

$\sqrt{44}$

Prove by the method of contradiction that each of the numbers given below is an irrational number.

$2\sqrt{3}$

Prove by the method of contradiction that each of the numbers given below is an irrational number.

$\sqrt{7}$

Prove by the method of contradiction that each of the numbers given below is an irrational number.

$\sqrt{2}$

Prove by the method contradiction that the product of a rational number and an irrational number is an irrational number.

Check whether the statement given below is true by the method of contradiction.
‘For a real number n, if $\mathrm{n} >4$ then ${\mathrm{n}}^2> 16$.’

Prove that the statement ‘For three integers $\mathrm{a}, \mathrm{b}, \mathrm{c},$ if $(\mathrm{a} - 2\mathrm{b})$ and $(2\mathrm{b}- \mathrm{c})$ both are divisible by $5$ then the number $(\mathrm{a} - \mathrm{c})$ is also divisible by $5$’, is true by $[\mathrm{il}$ direct method, $\left[\mathrm{ii}\right]$ method of contrapositive and $\left[\mathrm{iii}\right]$ method of contradiction.

$\mathrm{p} :$ If * is such a real number that ${\mathrm{x}}^3=-9\mathrm{x}$, then $\mathrm{x}=0$. Apply $\left[\mathrm{i}\right]$ direct method, $\left[\mathrm{ii}\right]$ method of contradiction$\left[\mathrm{iii}\right]$ method of contrapositive to prove that the above statement $\mathrm{p}$ is true.