Validating Statements

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

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

$~\left(p\to q\right)\wedge \left(~p\wedge q\right)$ is

The statement $~\left(p\to q\right)\leftrightarrow \left(~p\vee ~q\right)$ is

Let $p:57$ is an odd prime number.
$q:8$ is divisor of $24.$
$r:12$ is L.C.M of $3$ and $5$
be three simple logical statements. Which one of the following is not true?

Given below are two statements :

$p :$ $25$ is a multiple of $5$.

$q :$ $25$ is a multiple of $8$.

Write the compound statements connecting these two statements with ‘AND’ and ‘OR’. In both cases check the validity of the compound statement.

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

$100$ is a multiple of $4$ and $5$ is a statement.

 "Oh! it is too hot" is a statement.

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

Let $p:$ Maths is interesting and $q:$ Maths is easy, then $p\Rightarrow \left(~p\vee q\right)$ is equivalent to

Let $p:$ Maths is interesting and $q:$ Maths is easy, then $p\Rightarrow \left(~p\vee q\right)$ is equivalent to

Without using truth table, prove that $\left[\left(p\vee q\right)\wedge ~p\right]\to q$ is a tautology.

Using truth tables, prove the following logical equivalences:

$\left(p\wedge q\right)\to r\equiv p\to \left(q\to r\right)$

Using truth tables, prove the following logical equivalences:

$p\leftrightarrow q\equiv \left(p\wedge q\right)\vee \left(~p\wedge ~q\right)$

Using the rules of logic, prove the following logical equivalences.

$~p\wedge q\equiv \left(p\vee q\right)\wedge ~p$

Using the rules of logic, prove the following logical equivalences. 

$~\left(p\vee q\right)\vee \left(~p\wedge q\right)\equiv ~p$

Using truth tables, prove the following logical equivalences: 

$p\to q\equiv ~\left(p\wedge ~q\right)\wedge ~\left(q\wedge ~p\right)$

Determine whether the following statement patterns are tautologies, contradictions or contingencies.

$\left[\left(p\vee ~q\right)\vee \left(~p\wedge q\right)\right]\wedge r$

Determine whether the following statement patterns are tautologies, contradictions or contingencies.

$\left[p\wedge \left(p\to q\right)\right]\to q$