Q.Taking the set of natural numbers as the universal set, write down the complements of the following sets:

Concept: Set Membership — The complement of a set $A$ with respect to the universal set $U$ (here, $\mathbb{N}$) is $A' = U - A$, i.e., all elements of $\mathbb{N}$ that are not in $A$.

Reasoning steps:

1. Identify the given set $A$ in each case.
2. Remove all elements of $A$ from $\mathbb{N} = \{1,2,3,\dots\}$.
3. Write the remaining elements in set-builder form.

Concept First: Set Membership and Complements

A complement is simply the opposite of membership. If the universal set is $\mathbb{N}$, then for any set $A$, its complement $A'$ is:

So the trick is: read the condition that defines $A$, then write the condition that negates it. For example, if $A$ is "even natural numbers", then $A'$ is "natural numbers that are not even" — which means odd numbers.

Let's go through each one.

Step-by-Step Solutions

1. $A = \{x : x \text{ is an even natural number}\}$

Even numbers are $2,4,6,\dots$. The complement is all natural numbers that are not even — that is, all odd natural numbers.

$A' = \{x : x \text{ is an odd natural number}\}$

2. $B = \{x : x \text{ is an odd natural number}\}$

Odd numbers are $1,3,5,\dots$. The complement is all natural numbers that are not odd — that is, all even natural numbers.

$B' = \{x : x \text{ is an even natural number}\}$

3. $C = \{x : x \text{ is a positive multiple of 3}\}$

Multiples of 3: $3,6,9,12,\dots$. The complement is all natural numbers that are not multiples of 3.

$C' = \{x : x \in \mathbb{N} \text{ and } x \text{ is not a multiple of 3}\}$

4. $D = \{x : x \text{ is a prime number}\}$

Primes: $2,3,5,7,11,\dots$. The complement is all natural numbers that are not prime — that is, 1 and all composite numbers.

$D' = \{x : x \in \mathbb{N} \text{ and } x \text{ is either 1 or composite}\}$

5. $E = \{x : x \text{ is a natural number divisible by 3 and 5}\}$

"Divisible by 3 and 5" means divisible by LCM(3,5) = 15. So $E = \{15,30,45,\dots\}$. The complement is all natural numbers not divisible by 15.

$E' = \{x : x \in \mathbb{N} \text{ and } x \text{ is not divisible by 15}\}$

6. $F = \{x : x \text{ is a perfect square}\}$

Perfect squares: $1,4,9,16,25,\dots$. The complement is all natural numbers that are not perfect squares.

$F' = \{x : x \in \mathbb{N} \text{ and } x \text{ is not a perfect square}\}$

7. $G = \{x : x \text{ is a perfect cube}\}$

Perfect cubes: $1,8,27,64,\dots$. The complement is all natural numbers that are not perfect cubes.

$G' = \{x : x \in \mathbb{N} \text{ and } x \text{ is not a perfect cube}\}$

8. $H = \{x : x + 5 = 8\}$

Solve: $x + 5 = 8 \implies x = 3$. So $H = \{3\}$. The complement is all natural numbers except 3.

$H' = \{x : x \in \mathbb{N} \text{ and } x \neq 3\}$

9. $I = \{x : 2x + 5 = 9\}$

Solve: $2x + 5 = 9 \implies 2x = 4 \implies x = 2$. So $I = \{2\}$. The complement is all natural numbers except 2.

$I' = \{x : x \in \mathbb{N} \text{ and } x \neq 2\}$

10. $J = \{x : x \geq 7\}$

This set is $\{7,8,9,10,\dots\}$. The complement is all natural numbers that are less than 7.

$J' = \{x : x \in \mathbb{N} \text{ and } x < 7\} = \{1,2,3,4,5,6\}$

11. $K = \{x : x \in \mathbb{N} \text{ and } 2x + 1 > 10\}$

Solve: $2x + 1 > 10 \implies 2x > 9 \implies x > 4.5$. Since $x$ is a natural number, $x \geq 5$. So $K = \{5,6,7,8,\dots\}$. The complement is all natural numbers that are less than or equal to 4.

$K' = \{x : x \in \mathbb{N} \text{ and } x \leq 4\} = \{1,2,3,4\}$

Method: Complement of a Set Using the Universal Set

Method Name: Universal Set Complement Method

Concept: The complement of a set $A$ (denoted $A'$ or $\overline{A}$) is the set of all elements in the universal set $U$ that are not in $A$.

Steps

1. Identify the universal set — here, $U = \mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$
2. Write the given set in roster or set-builder form
3. Remove all elements of the given set from $\mathbb{N}$
4. Express the complement in set-builder notation

Solutions

(i) $\{x : x \text{ is an even natural number}\}$

Complement = $\{x : x \text{ is an odd natural number}\}$

(ii) $\{x : x \text{ is an odd natural number}\}$

Complement = $\{x : x \text{ is an even natural number}\}$

(iii) $\{x : x \text{ is a positive multiple of 3}\}$

Complement = $\{x : x \in \mathbb{N}, x \text{ is not a multiple of 3}\}$

(iv) $\{x : x \text{ is a prime number}\}$

Complement = $\{x : x \in \mathbb{N}, x = 1 \text{ or } x \text{ is composite}\}$

(v) $\{x : x \text{ is divisible by 3 and 5}\}$

Since divisible by both means divisible by LCM = 15,

Complement = $\{x : x \in \mathbb{N}, x \text{ is not divisible by 15}\}$

(vi) $\{x : x \text{ is a perfect square}\}$

Complement = $\{x : x \in \mathbb{N}, x \text{ is not a perfect square}\}$

(vii) $\{x : x \text{ is a perfect cube}\}$

Complement = $\{x : x \in \mathbb{N}, x \text{ is not a perfect cube}\}$

(viii) $\{x : x + 5 = 8\}$

Solving: $x = 3$, so set = $\{3\}$

Complement = $\mathbb{N} - \{3\} = \{x : x \in \mathbb{N}, x \neq 3\}$

(ix) $\{x : 2x + 5 = 9\}$

Solving: $2x = 4 \Rightarrow x = 2$, so set = $\{2\}$

Complement = $\mathbb{N} - \{2\} = \{x : x \in \mathbb{N}, x \neq 2\}$

(x) $\{x : x \geq 7\}$

Complement = $\{x : x \in \mathbb{N}, x < 7\} = \{1, 2, 3, 4, 5, 6\}$

(xi) $\{x : x \in \mathbb{N} \text{ and } 2x + 1 > 10\}$

Solving: $2x > 9 \Rightarrow x > 4.5$, so $x \geq 5$

Complement = $\{x : x \in \mathbb{N}, x \leq 4\} = \{1, 2, 3, 4\}$

Key Exam Tip: Always check whether the given set includes $1$ (since $1$ is neither prime nor composite) and whether natural numbers start from $1$ (as per NCERT convention).

This is a classic from NCERT Class 11 (Chapter 1: Sets). Let’s go through the common mistakes students make and how to avoid each.

🧠 Mistake 1: Forgetting the Universal Set

The mistake:

Students write complements that include numbers outside $\mathbb{N}$ (like $0$, negative numbers, or fractions).

Example:

For (i) $\{x : x \text{ is an even natural number}\}$, a student writes the complement as $\{x : x \text{ is an odd integer}\}$ — which includes negative odd numbers and zero.

Why it’s wrong:

The universal set is $\mathbb{N} = \{1, 2, 3, \dots\}$. The complement must be a subset of $\mathbb{N}$.

How to avoid:

Always first write down the universal set explicitly. Then, for any complement, ask: “Is every element I’m listing actually in $\mathbb{N}$?”

✓ Correct complement for (i):

$\{x : x \in \mathbb{N} \text{ and } x \text{ is odd}\}$

🧠 Mistake 2: Including 0 in $\mathbb{N}$

The mistake:

Assuming $\mathbb{N}$ includes $0$ (common in some textbooks, but not in NCERT for this chapter).

Example:

For (ii) $\{x : x \text{ is an odd natural number}\}$, a student writes complement as $\{0, 2, 4, 6, \dots\}$.

Why it’s wrong:

NCERT defines $\mathbb{N} = \{1, 2, 3, \dots\}$. $0$ is not a natural number here.

How to avoid:

Check the definition used in your syllabus. For NCERT Class 11, $\mathbb{N}$ starts at 1.

✓ Correct complement for (ii):

$\{x : x \in \mathbb{N} \text{ and } x \text{ is even}\}$

🧠 Mistake 3: Confusing “multiple of 3” with “divisible by 3”

The mistake:

Treating (iii) and (v) as the same.

Why they differ:

• (iii) $\{x : x \text{ is a positive multiple of } 3\} = \{3, 6, 9, 12, \dots\}$
• (v) $\{x : x \text{ is divisible by } 3 \text{ and } 5\} = \{x : x \text{ is a multiple of } 15\}$

How to avoid:

Read carefully: “multiple of 3” means $3, 6, 9, \dots$; “divisible by 3 and 5” means divisible by LCM = 15.

✓ Complement of (iii):

$\{x : x \in \mathbb{N}, x \text{ not a multiple of } 3\}$

✓ Complement of (v):

$\{x : x \in \mathbb{N}, x \text{ not divisible by } 15\}$

🧠 Mistake 4: Forgetting that 1 is not prime

The mistake:

In (iv), writing complement of primes as $\{1, 4, 6, 8, 9, \dots\}$ — but forgetting that $1$ is not prime and not composite.

Why it’s wrong:

$1$ is a natural number that is neither prime nor composite. It belongs to the complement.

How to avoid:

Memorise: 1 is not prime. Always list it separately when writing complements of primes.

✓ Correct complement for (iv):

$\{1, 4, 6, 8, 9, 10, 12, \dots\}$ (all natural numbers except primes)

🧠 Mistake 5: Solving equations incorrectly

The mistake:

For (viii) $\{x : x + 5 = 8\}$, a student writes complement as $\{x : x \neq 3\}$ — but forgets that $x$ must be in $\mathbb{N}$.

Why it’s wrong:

The set itself is $\{3\}$. Its complement in $\mathbb{N}$ is $\mathbb{N} \setminus \{3\}$, i.e., all natural numbers except 3.

How to avoid:

First solve the equation within $\mathbb{N}$, then subtract that single element from $\mathbb{N}$.

✓ Correct complement for (viii):

$\{x : x \in \mathbb{N}, x \neq 3\}$

🧠 Mistake 6: Misinterpreting inequalities

The mistake:

For (x) $\{x : x \geq 7\}$, a student writes complement as $\{x : x < 7\}$ — but forgets $x \in \mathbb{N}$.

Why it’s wrong:

In $\mathbb{N}$, $x < 7$ means $x \in \{1, 2, 3, 4, 5, 6\}$. That’s correct — but the mistake is writing it as $\{x : x < 7\}$ without specifying $x \in \mathbb{N}$.

How to avoid:

Always specify the universal set in the complement description.

✓ Correct complement for (x):

$\{x : x \in \mathbb{N}, x < 7\}$ or simply $\{1, 2, 3, 4, 5, 6\}$

🧠 Mistake 7: Forgetting to list elements for small sets

The mistake:

For (ix) $\{x : 2x + 5 = 9\}$, a student writes complement as $\{x : x \neq 2\}$ — but doesn’t check if $2$ is in $\mathbb{N}$.

Why it’s wrong:

$2$ is in $\mathbb{N}$, so the complement is $\mathbb{N} \setminus \{2\}$. But writing it as $\{x : x \neq 2\}$ is acceptable only if you state $x \in \mathbb{N}$.

How to avoid:

For small finite sets, list the elements explicitly — it’s safer.

✓ Correct complement for (ix):

$\{1, 3, 4, 5, 6, \dots\}$ or $\{x : x \in \mathbb{N}, x \neq 2\}$

🧠 Mistake 8: Confusing “perfect square” with “square number”

The mistake:

For (vi), a student writes complement as $\{2, 3, 5, 6, 7, 8, 10, \dots\}$ — but forgets that $1$ is a perfect square ($1^2 = 1$).

Why it’s wrong:

$1$ is a perfect square, so it should be excluded from the complement.

How to avoid:

List the first few perfect squares: $1, 4, 9, 16, \dots$ — then remove them from $\mathbb{N}$.

✓ Correct complement for (vi):

$\{2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, \dots\}$

🧠 Mistake 9: Forgetting that 1 is also a perfect cube

The mistake:

For (vii), a student writes complement as $\{2, 3, 4, 5, 6, 7, 9, \dots\}$ — forgetting $1 = 1^3$.

How to avoid:

Same as above — list cubes: $1, 8, 27, 64, \dots$

✓ Correct complement for (vii):

$\{2, 3, 4, 5, 6, 7, 9, 10, \dots\}$ (all naturals except perfect cubes)

🧠 Mistake 10: Writing complements in set-builder form without the universal set

The mistake:

Writing complement of (xi) $\{x : x \in \mathbb{N}, 2x + 1 > 10\}$ as $\{x : 2x + 1 \leq 10\}$ — without specifying $x \in \mathbb{N}$.

Why it’s wrong:

The complement must be a subset of $\mathbb{N}$. The condition $2x + 1 \leq 10$ gives $x \leq 4.5$, so $x \in \{1, 2, 3, 4\}$.

How to avoid:

Always solve the inequality and then list or describe the resulting natural numbers.

✓ Correct complement for (xi):

$\{1, 2, 3, 4\}$ or $\{x : x \in \mathbb{N}, x \leq 4\}$

✓ Final Quick Checklist