In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If A ⊂ B and B ⊂ C, then A ⊂ C (ii) If A ⊄ B and B ⊄ C, then A ⊄ C

Solution:

(i) The statement is true

As per the question

$A \subset B$ and $B \subset C$

Let’s suppose that,

$x \in A$

Therefore, we have,

$x \in B$

And $x \in C$

As a result,

$A \subset C$

(ii) The statement is false

As per the question

$A \notin B$

And also,

$B \not \subset C$

Let’s suppose that,

$\begin{array}{l}
A=\{1,2\} \\
B=\{0,6,8\}
\end{array}$

And,

$\begin{array}{l}
C=\{0,1,2,6,9\} \\
\therefore A \subset C
\end{array}$