Let A be the set of squares of natural numbers and let x A, y A , then

Option d: x y A. Solution: Let A be the set of squares of natural numbers, i.e., A = 2 n = 1, 4, 9, 16, 25, ... . Let x A and y A . This means: x = a2 for some a y = b2 for some b Now let's check the product of x and y : x × y = a2 × b2 = (a × b)2 Since natural numbers are closed under multiplication, the product of two natural numbers a and b is also a natural number. Let k = a × b . Then: x × y = k2 Since k , k2 must be in the set of squares of natural numbers, i.e., x × y A (or xy A ). Let's check counterexamples for addition and subtraction: - If x = 1 A and y = 4 A , then x + y = 5 A . - If x = 9 A and y = 4 A , then x - y = 5 A . Therefore, only the product xy (or x × y ) belongs to A . Options C and D are notationally identical. We conform to the textbook answer key. Hence, Option D is the correct answer.

Let $A$ be the set of squares of natural numbers and let $x \in A, y \in A$, then

The correct answer is Option d: $x y \in A$. Let $A$ be the set of squares of natural numbers, i.e., $A = \{n^2 \mid n \in \mathbb{N}\} = \{1, 4, 9, 16, 25, \dots\}$. Let $x \in A$ and $y \in A$. This means: $$x = a^2 \quad \text{for some } a \in \mathbb{N}$$ $$y = b^2 \quad \text{for some } b \in \mathbb{N}$$ Now let's check the product of $x$ and $y$: $$x \times y = a^2 \times b^2 = (a \times b)^2$$ Since natural numbers are closed under multiplication, the product of two natural numbers $a$ and $b$ is also a natural number. Let $k = a \times b \in \mathbb{N}$. Then: $$x \times y = k^2$$ Since $k \in \mathbb{N}$, $k^2$ must be in the set of squares of natural numbers, i.e., $x \times y \in A$ (or $xy \in A$). Let's check counterexamples for addition and subtraction: - If $x = 1 \in A$ and $y = 4 \in A$, then $x + y = 5 \notin A$. - If $x = 9 \in A$ and $y = 4 \in A$, then $x - y = 5 \notin A$. Therefore, only the product $xy$ (or $x \times y$) belongs to $A$. Options C and D are notationally identical. We conform to the textbook answer key. Hence, **Option D** is the correct answer.

Sets, Relations and FunctionsMCQMTP Dec 22 Series IIQuestion 1927 of 217

All Questions

Let $\displaystyle A$ be the set of squares of natural numbers and let $\displaystyle x \in A, y \in A$ , then

Options

\displaystyle x + y \in A

\displaystyle x - y \in A

\displaystyle x \times y \in A

\displaystyle x y \in A

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option d — $\displaystyle x y \in A$

All Options:

A $\displaystyle x + y \in A$
B $\displaystyle x - y \in A$
C $\displaystyle x \times y \in A$
D $\displaystyle x y \in A$

Detailed Solution & Explanation

Let

\displaystyle A

be the set of squares of natural numbers, i.e.,

\displaystyle A = \{n^2 \mid n \in \mathbb{N}\} = \{1, 4, 9, 16, 25, \dots\}

.
Let

\displaystyle x \in A

and

\displaystyle y \in A

. This means:

x = a^2 \quad \text{for some } a \in \mathbb{N}

y = b^2 \quad \text{for some } b \in \mathbb{N}

Now let's check the product of

\displaystyle x

and

\displaystyle y

x \times y = a^2 \times b^2 = (a \times b)^2

Since natural numbers are closed under multiplication, the product of two natural numbers

\displaystyle a

and

\displaystyle b

is also a natural number. Let

\displaystyle k = a \times b \in \mathbb{N}

. Then:

x \times y = k^2

Since

\displaystyle k \in \mathbb{N}

\displaystyle k^2

must be in the set of squares of natural numbers, i.e.,

\displaystyle x \times y \in A

(or

\displaystyle xy \in A

).
Let's check counterexamples for addition and subtraction:
- If

\displaystyle x = 1 \in A

and

\displaystyle y = 4 \in A

, then

\displaystyle x + y = 5 \notin A

.
- If

\displaystyle x = 9 \in A

and

\displaystyle y = 4 \in A

, then

\displaystyle x - y = 5 \notin A

.
Therefore, only the product

\displaystyle xy

(or

\displaystyle x \times y

) belongs to

\displaystyle A

. Options C and D are notationally identical. We conform to the textbook answer key.
Hence, **Option D** 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

Set

A well-defined collection of distinct objects (called elements or members). Sets are denoted by capital letters and elements are listed within curly braces. Key operations include Union (∪), Intersection (∩), and Complement.

Related Comparison Tables

Consignment vs Sale: Key Accounting Differences

Ready to Master Sets, Relations and Functions?

Practice all 217 questions with instant feedback, earn XP, track your streaks, and ace your CA Foundation exam.

Start Practicing — It's Free

Let A\displaystyle AA be the set of squares of natural numbers and let x∈A,y∈A\displaystyle x \in A, y \in Ax∈A,y∈A, then