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 $A\subset C$

(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) The statement is false.

Take A = {1}, B = {{1}, 2}

Now $1\in A$ and $A\in B$ but $1\notin B.$

(ii) The statement is false

Take A = {1}, B = {1, 2}, C = {{1, 2}, 3}

Now $A\subset B$ and $B\in C$ but $A\notin C.$

(iii) The statement is true.

Let $x\in A\Rightarrow x\in B$ $(\because A\subset B)$

$\Rightarrow x\in C$ $(\because B\subset C)$

Now $x\in A\Rightarrow x\in C$

$\therefore A\subset C.$

(iv) The statement is false.

Take A = {1, 2} B = {2, 3 }, C = {1, 2, 5}

Now $A\subset ̸B$ and $B\subset ̸C$ but $A\subset C.$

(v) The statement is false.

Take A = {1, 2} and B = {2, 3, 4, 5}

Now $1\in A$ and $A\subset ̸B$ but $1\notin B$

(vi) The statement is true.

Let $x\in A\Rightarrow x\in B$ $(\because A\subset B)$

Now $x\notin B\Rightarrow x\notin A.$