Name: Sets: Intersection and Union
Uploaded: 2019-07-19
Duration: 5 min 51 s
Description: Suppose you have a Venn diagram but want to find out the intersection of sets and the union of sets. Learn more by watching this video.

Question

A survey was conducted to know the hobby of $220$ students of class IX. Out of which $130$ students informed about their hobby as rock climbing and $180$ students informed about their hobby as sky watching. There are $110$ students who follow both the hobbies. Then how many students do not have any of the two hobbies? How many of them follow the hobby of rock climbing only? How many students follow the hobby of sky watching only?

Accepted Answer

Given: $n (A) = 130, n (B) = 180$ and $n (A \cap B) = 110$

Total number of students $= 220$

Consider,

$n (A) =$ Number of students whose hobby is rock climbing

$n (B) =$ Number of students whose hobby is sky watching

$n (A \cap B) =$ Number of students who follow both the hobbies

$n (A \cup B) =$ Number of students whose hobby is either rock climbing or sky watching

We know that,

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$

$\Rightarrow n (A \cup B) = 130 + 180 - 110$

$\Rightarrow n (A \cup B) = 200$

So, the number of students who do not have any hobby = total number of students $- n (A \cup B)$

$= 220 - 200$ $= 20$

Hence, $20$ students do not have any of the two hobbies.

Now, the number of students whose hobby is rock climbing only

$= n (A) - n (A \cap B)$

$= 130 - 110$

$= 20$

Hence, $20$ students follow the hobby of rock climbing only.

Finally, the number of students whose hobby is sky watching only

$= n (B) - n (A \cap B)$

$= 180 - 110$

$= 70$

Hence, $70$ students follow the hobby of sky watching only.

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