Sets & RelationsmediumFree

Find n from Overlapping Sets A and B Membership Counts | JEE

JEE Maths question with a full step-by-step solution.

Question
Suppose A1,A2,…,A30A_1, A_2, \ldots, A_{30} are thirty sets each having 55 elements and B1,B2,…,BnB_1, B_2, \ldots, B_n are nn sets each with 33 elements. Let ⋃i=130Ai=⋃j=1nBj=S\bigcup_{i=1}^{30} A_i = \bigcup_{j=1}^{n} B_j = S, and each element of SS belongs to exactly 1010 of the AiA_i and exactly 99 of the BjB_j. Then nn is equal to:
A1515
B33
C4545correct
D3535
Solution
Step 1: Count memberships through the AiA_i. Total =30×5=150= 30 \times 5 = 150, and each element of SS is counted 1010 times:
10⋅n(S)=150⇒n(S)=15.10 \cdot n(S) = 150 \Rightarrow n(S) = 15.
 Step 2: Count memberships through the BjB_j. Total =3n= 3n, and each element of SS is counted 99 times:
3n=9⋅n(S)=9×15=135⇒n=45.3n = 9 \cdot n(S) = 9 \times 15 = 135 \Rightarrow n = 45.
 Correct answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Sets & Relations · easy
The relation R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\} on the set A={1,2,3}A = \{1, 2, 3\} is:
Sets & Relations · easy
The relation RR is defined in A={1,2,3}A = \{1, 2, 3\} by aRbaRb if ∣a2−b2∣≤5|a^2 - b^2| \le 5. Which of the following is false?
Sets & Relations · medium
The relation RR defined on the set A={1,2,3,4,5}A = \{1, 2, 3, 4, 5\} by R={(x,y):∣x2−y2∣<16}R = \{(x, y) : |x^2 - y^2| < 16\} is given by:
Sets & Relations · easy
If AA is the set of even natural numbers less than 88 and BB is the set of prime numbers less than 77, then the number of relations from AA to BB is:

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath