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Question

If $A$ and $B$ are two sets such that $n open parentheses A close parentheses equals 7 comma n open parentheses B close parentheses equals 6$ and $open parentheses A intersection B close parentheses not equal to ϕ.$ Then the greatest possible value of $n open parentheses A blank capital delta blank B close parentheses comma$ is

(a) 11
12
13
10

The correct answer is: (a) 11

We have,
$A blank increment blank B equals open parentheses A union B close parentheses minus left parenthesis A union B right parenthesis$
$rightwards double arrow n open parentheses A blank increment blank B close parentheses equals n open parentheses A close parentheses plus n open parentheses B close parentheses minus 2 blank n left parenthesis A intersection B right parenthesis$
So, $n left parenthesis A blank increment blank B right parenthesis$ is greatest when $n left parenthesis A intersection B right parenthesis$ is least
It is given that $A intersection B not equal to ϕ.$ So, least number of elements in $A blank intersection B$ can be one
$therefore$ Greatest possible value of $n left parenthesis A blank increment blank B right parenthesis$ is $7 plus 6 minus 2 cross times 1 equals 11$

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