Question

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 $x\in A$ and $A\in B$, then $x\in B$
(ii) If $A\subset B$ and $B\in C$, then $A\in C$
(iii) If $A\subset B$ and $B\subset C$, then $AC$
(iv) If $A\subset ̸B$ and $B\subset ̸C$, then $A\subset ̸C$
(v) If $x\in A$ and $A\subset ̸B$, then $x\in B$
(vi) If $A\subset B$ and $x\notin B$, then $x\notin A$

Answer 1

(i) False

Let $A=\{1,2\}$ and $B=\{1,\{1,2\},\left\{3\right\}\}$

$2\in \{1,2\}$ and $\{1,2\}\in \{\left\{3\right\},1,\{1,2\}\}$

Now,

$\therefore A\in B$

However, $2\text{/}\in \{\left\{3\right\},1,\{1,2\}\}$

(ii) False.

As $A\subset B$, $B\in C$

Let $A=\left\{2\right\},B=\{0,2\}$, and $C=\{1,\{0,2\},3\}$

However, $A\text{/}\in C$

(iii) True

Let $A\subset B$ and $B\subset C$.

Let $x\in A$

$\Rightarrow x\in B$ $[\because A\subset B]$

$\Rightarrow x\in C$ $[\because B\subset C]$

$\therefore A\subset C$

(iv) False

As, $A\text{/}\subset B$ and $B\text{/}\subset C$

Let $A=\{1,2\},B=\{0,6,8\}$, and $C=\{0,1,2,6,9\}$

However, $A\subset C$

(v) False

Let $A=\{3,5,7\}$ and $B=\{3,4,6\}$

Now, $5\in A$ and $A\text{/}\subset B$

However, $5\text{/}\in B$

(vi) True

Let $A\subset B$ and $x\text{/}\in B$.

To show: $x\text{/}\in A$

If possible, suppose $x\in A$.

Then, $x\in B$, which is a contradiction as $x\text{/}\in B$

$\therefore x\text{/}\in A$