Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

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

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

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

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

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

Examine whether each of the following statement patterns is a tautology or a contradiction or a contingency.

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

If $A=\left\{3,5,7,9,11,12\right\}$, determine the truth value of each of the following.

$\exists x\in A$,    such that $x-8=1$

If $A=\left\{3,5,7,9,11,12\right\}$, determine the truth value of each of the following.

$\forall x\in A,x^2+x$ is an even number

If $A=\left\{3,5,7,9,11,12\right\}$, determine the truth value of each of the following.

$\exists x\in A$ such that $x^2<0$

If $A=\left\{3,5,7,9,11,12\right\}$, determine the truth value of each of the following.

$\forall x\in A, x$ is an even number

If $A=\left\{3,5,7,9,11,12\right\}$, determine the truth value of each of the following.

$\exists x\in A$ such that $3x+8>40$