Sets, Relations and FunctionsMCQPYQ June 22Question 1897 of 217
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If the universal set E={xx is a positive integer <25}\displaystyle E = \{x \mid x \text{ is a positive integer } < 25\}, A={2,6,8,14,22}\displaystyle A = \{2, 6, 8, 14, 22\}, B={4,8,10,14}\displaystyle B = \{4, 8, 10, 14\}

Options

A(AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cup B^c
B(AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cap B^c
C(AB)c=ϕ\displaystyle (A - B)^c = \phi
DNone of these
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Correct Answer

Option a(AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cup B^c

All Options:

  • A(AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cup B^c
  • B(AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cap B^c
  • C(AB)c=ϕ\displaystyle (A - B)^c = \phi
  • DNone of these

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Detailed Solution & Explanation

We are given:
Universal set E={1,2,3,,24}\displaystyle E = \{1, 2, 3, \dots, 24\}
Subsets A={2,6,8,14,22}\displaystyle A = \{2, 6, 8, 14, 22\} and B={4,8,10,14}\displaystyle B = \{4, 8, 10, 14\}
Let us evaluate Option A: (AB)c=AcBc\displaystyle (A \cap B)^c = A^c \cup B^c.
By **De Morgan's Laws** in set theory, for any two sets A\displaystyle A and B\displaystyle B within a universal set E\displaystyle E, the complement of their intersection is the union of their complements:
(AB)c=AcBc(A \cap B)^c = A^c \cup B^c
Let us verify this explicitly:
- Intersection: AB={8,14}\displaystyle A \cap B = \{8, 14\}
- Complement of intersection: (AB)c=E{8,14}\displaystyle (A \cap B)^c = E - \{8, 14\}
- Complement of A\displaystyle A: Ac=E{2,6,8,14,22}\displaystyle A^c = E - \{2, 6, 8, 14, 22\}
- Complement of B\displaystyle B: Bc=E{4,8,10,14}\displaystyle B^c = E - \{4, 8, 10, 14\}
- Union of complements: AcBc=E(AB)=E{8,14}\displaystyle A^c \cup B^c = E - (A \cap B) = E - \{8, 14\}
Since both sides are identical, De Morgan's Law holds.
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.

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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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