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Question 8 of 31 · Set Theory
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NIMCET 2013

Find the number of elements in the union of 4 sets A, B, C and D having 150, 180, 210 and 240 elements respectively, given that each pair of sets has 15 elements in common, each triple of sets has 3 elements in common and .

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Answer: Option D — 702

Detailed Solution

1

Inclusion-exclusion for 4 sets: union = (sum of singles) - (sum of pairs) + (sum of triples) - (all four).

2

Singles: . There are pairs giving and triples giving ; the quadruple is 0.

3

Union .

✓ Check

The union must lie between 240 (the largest set) and 780 (the sum of sizes); 702 does.

⚠ Common trap

(C) 111 is smaller than set D alone (240 elements), so it is impossible; forgetting the +12 for triples gives 690, which is not listed.

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Set Theory — Preparation

This question tests your understanding of Set Theory, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A set is a well-defined collection of objects. Set theory covers types of sets (finite, infinite, empty, singleton, equal and equivalent sets), subsets and power sets, and operations on sets such as union, intersection, difference and complement. It also covers De Morgan's laws and the cardinality formula n(A union B) = n(A) + n(B) - n(A intersection B). In entrance exams, the most common questions are word problems about people who like one, two or three activities, solved with Venn diagrams and the cardinality formula. Questions also ask for the number of subsets of a set (2^n for a set with n elements) and test set notation and properties. Practise the set theory questions below. Each comes with a step-by-step solution and a Venn diagram where it helps.

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