Statements patterns and Logical Equivalence

The statement $(p\wedge (~q\left)\right)\vee \left(\right(~p) \wedge q) \vee \left(\right(~p) \wedge (~q\left)\right)$ is equivalent to $\_\_\_\_\_$

Among the statements

$\left(S1\right): \left(p\Rightarrow q\right)\vee \left(\left(~p\right)\wedge q\right)$ is a tautology

$\left(S2\right): \left(q\Rightarrow p\right)\Rightarrow \left(\left(~p\right)\wedge q\right)$ is a contradiction

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

Number of the form $2 n + 1$ where $n$ is any positive integer are always odd number.

By giving a counter example, show that the following statement is false. If $n$ is an odd integer, then $n$ is prime.

Verify by the method of contradiction.

$p:\sqrt{7}$ is irrational

Check whether the following statement is true or false by proving its contrapositive. If $x, y\in Z$ such that $xy$ is odd, then both $x$ and $y$ are odd.

Check whether the following statement is true or not. If $x, y\in Z$ are such that $x$ and $y$ are odd, then $xy$ is odd.

State whether the following statement is true or false.

The $P\to Q$ and $\neg P\vee Q$ relations are logically equivalent.

Show that $P\to Q$ and $\neg P\vee Q$ are logically equivalent.

Find if  $~A\wedge B \Rightarrow ~(A\vee B)$ is a tautology or not.

 Show that $p\Rightarrow (p\vee q)$ is a tautology.

Consider the statement:

$q:$ for any real number $a \& b$, $a^2=b^2\Rightarrow a=b$.

By giving a counterexample, prove that $q$ is false.

By giving a counterexample, show that the following statement is not true.

$q:$ The equation $x^2-1=0$ does not have a root lying between $0 \& 2$.

Check the validity of the statement given below by the method given against it.

$q:$ If $n$is a real number with $n>3,$ then $n^2>9$ (By contradiction method)

Show that the statement $p:$ If $x$ is a real number such that $x^3+4x=0,$ then $x$ is $0$ is true by the method of contradiction.

Show that the statement $p:$ If $x$ is a real number such that $x^3+4x=0,$ then $x$ is $0$ is true by the method of contrapositive.

Let $p:$If $x$ is an integer and $x^2$ is even, then $x$ is even. Using the method of contrapositive, prove that $p$ is true.

Check the validity of the compound statement using the direct method.

If $x$ is a number such that $2x^3+5x=0$, then $x=0.$

Check the validity of the compound statement using the direct method.

If $x$ is a number such that $4x^3+3x=0$, then $x=0.$