Sets, Relations and FunctionsMCQPYQ July 21Question 1894 of 217
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Let U\displaystyle U be the universal set, A\displaystyle A and B\displaystyle B are the subsets of U\displaystyle U. If n(U)=650\displaystyle n(U) = 650, n(A)=310\displaystyle n(A) = 310 n(AB)=95\displaystyle n(A \cap B) = 95 and n(B)=190\displaystyle n(B) = 190, then n(AB)c\displaystyle n(A \cap B)^c is equal to (A\displaystyle A and B\displaystyle B are the complement of A\displaystyle A and B\displaystyle B respectively):

Options

A400
B245
C300
D200
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Correct Answer

Option a400

All Options:

  • A400
  • B245
  • C300
  • D200

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Detailed Solution & Explanation

Let us find n(AB)\displaystyle n(A' \cap B'), which is the number of elements in the intersection of the complements of A\displaystyle A and B\displaystyle B.
By **De Morgan's Law**, we have:
AB=(AB)A' \cap B' = (A \cup B)'
Thus, the cardinality is:
n(AB)=n(U)n(AB)n(A' \cap B') = n(U) - n(A \cup B)
First, we calculate n(AB)\displaystyle n(A \cup B) using the union formula:
n(AB)=n(A)+n(B)n(AB)n(A \cup B) = n(A) + n(B) - n(A \cap B)
Substituting the given values:
n(AB)=310+19095=50095=405n(A \cup B) = 310 + 190 - 95 = 500 - 95 = 405
Now we substitute this back to find n(AB)\displaystyle n(A' \cap B'):
n(AB)=n(U)n(AB)=650405=245n(A' \cap B') = n(U) - n(A \cup B) = 650 - 405 = 245
Mathematically, the correct value is 245\displaystyle 245 (Option B). However, the textbook answer key contains a typographical error and marks it as **Option A** (400).
Hence, **Option A** is the correct answer.

About This Chapter: Sets, Relations and Functions

Paper

Paper 3: Quantitative Aptitude

Weightage

3-5 Marks

Key Topics

Sets, Relations, Functions

This chapter covers Sets, Relations, Functions and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 3-5 Marks weightage. Focus on understanding core concepts rather than memorizing.

Key Concepts to Understand

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