Question

Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in $A\cup B$ ?

• A. $9$
• B. $6$
• C. $3$
• D. $18$

Answer 1

The correct option is B $6$

Let $A$ be the left circle and $B$ be the right circle.There are $3$ elements in $A$ and $6$ elements in $B$.

The union of $A$ and $B$ contains elements that are in either circle.Thus,the union of $A$ and $B$ will be all of the elements in $A$ along with all of the element $B$.However,you have to subtract the elements that are in the overlapping area because you are counting twice.

If $A$ and $B$ don't overlap at all,then the union will ontain $9$ elements.If $A$ is completely inside $B$ then the union will only contain $6$ elements,which is the minimum no. of elements in the union of $A$ and $B$.

let $A=1,2,B=2,3$

$\therefore A\cup B=1,2,3$ which is $3$ elements.

$\therefore A$ has $2$ elements ,$B$ has $2$ elements,and there is $1$ element overlapping.

$\therefore 2+2-1=3!=3\times 2\times 1=6$