Determine whether the statement given below is true or false. If it is true, prove it. If it is false, give an example:

If $\mathrm{A}\subset \mathrm{B} \mathrm{and} \mathrm{B}\in \mathrm{C},$ then $A\in C$.

Determine whether the statement given below is true or false. If it is true, prove it. If it is false, give an example:

If $\mathrm{A}\subset \mathrm{B} \mathrm{and} \mathrm{B}\subset \mathrm{C},$ then $A\subset C$.

Determine whether the statement given below is true or false. If it is true, prove it. If it is false, give an example:

If $\mathrm{A}\not\subset \mathrm{B} \mathrm{and} \mathrm{B}\not\subset \mathrm{C},$ then $A\not\subset C$.

Determine whether the statement given below is true or false. If it is true, prove it. If it is false, give an example:

If $\mathrm{x}\in \mathrm{A} \mathrm{and} A\not\subset B,$ then $x\in B$.

Determine whether the statement given below is true or false. If it is true, prove it. If it is false, give an example:

If $\mathrm{A}\subset \mathrm{B} \mathrm{and} x\notin B,$ then $x\notin A$.

If $A=6^n-5n-1, n\in N$ and $B=25\left(n-1\right)$, $n\in N$ then prove that $A\subseteq B$.

Write down the following statement in set theoretic notation:

$A$ is a subset of $B$.

Write down the following statement in set theoretic notation:

$C$ is a proper subset of $B$.

Write down the following statement in set theoretic notation:

$C$ and $D$ are disjoint sets.