Question

Determine which of the following are true?
I. If $x∈A$ and $A∈B$, then $x∈B$.
II. If $A⊂B$ and $B∈C$, then $A∈C$.
III. If $A⊂B$ and $B⊂C$, then $A⊂C$.
IV. If $Aî€âŠ‚B$ and $Bî€âŠ‚C$, then $Aî€âŠ‚C$.
V. If $x∈A$ and $Aî€âŠ‚B$, then $x∈B$.
VI. If $A⊂B$ and $x∈/B$, then $x∈/A$.

• A. I, IV and VI
• B. III, VI
• C. II , III and V
• D. All of these

Answer 1

Answer: (B)

I. If $x∈A$ and $A∈B⇒x∈/B$
As $A=\{1,2\},B=\{\{1,2\},3\}$
Then, $1∈A$ but $1∈/B$
II. If $A⊂B$ and $B∈C⇒A∈/C$
As $A=\{1\},B=\{1,2\}$ and $C=\{\{1,2\},3\}$
III. If $A⊂B$ and $B⊂C$
$⇒$ Every element of $A$ is in $B$ and every element of $B$ is in $C$
$⇒$ Every element of $A$ is also in $C$.
$⇒A⊂C$
IV. If $Aî€âŠ‚B$ and $Bî€âŠ‚C$, then $A$ may be subset of $C$.
As $A=\{1,2\},B=\{3\},C=\{1,2,4,5\}$
V. If $x∈A$ and $Aî€âŠ‚B⇒x∈/B$
As $A=\{1,2\},B=\{3,4,5\}$
Here, $1∈A$ and $Aî€âŠ‚B$ but $1∈/B$
VI. If $A⊂B$ and $x∈/B$, then $x∈/A$
As $A⊂B$
$⇒$ Every element of $A$ is in $B$.
i.e., there does not exist any element of $A$ which does not belong to $B$.