Single Choice

If $$A=\left\{ 1,2,3,4,5,6 \right\} $$, $$B=\left\{ 1,2 \right\} $$, then $$ \displaystyle \dfrac { A }{ \left( \dfrac { A }{ B }\right)} $$ is equal to

A$$A, B$$
B$$B, $$ $$A\cup B$$
C$$A\cap B, B$$
Correct Answer
D$$A\cup B$$

Solution

Difference of two sets is given by $$\dfrac{A}{B}$$ or $$A-B$$. means all those elements of Set $$A$$ which is not present in set $$B$$.
$$\therefore \dfrac{A}{B}=A-B=\left \{ 3,4,5,6 \right \}$$
Now, $$\dfrac{A}{\left(\dfrac{A}{B}\right)}$$ means all those elements of set $$A$$ which is not present in the set obtained above or in the set $${A}$$.
$$\therefore \dfrac{A}{\left(\dfrac{A}{B}\right)}=\left \{ 1,2 \right \}$$
$$A\cap B=\left \{1,2 \right \}$$\therefore A\cap B=\dfrac{A}{\left(\dfrac{A}{B}\right)}=\left \{1,2 \right \}$$.
Also, note that $$A\cap B=\left \{1,2 \right \} =B$$.
Thus, correct answer for this question is $$A\cap B, B$$.

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