Q.Which of the following are examples of the null set

Concept: Null Set Identification — A set with no elements is called a null set (or empty set), denoted by $\emptyset$.

Reasoning:

1. (i) Odd numbers are not divisible by 2, so no odd number can be divisible by 2. Hence this set has no elements.
2. (ii) The only even prime number is 2, so this set contains one element — it is not empty.
3. (iii) No natural number can be simultaneously less than 5 and greater than 7. The condition is impossible.
4. (iv) Parallel lines never intersect, so they share no common point.

Thus, the null sets are (i), (iii), and (iv).

The idea of a null set (or empty set) is simple: it’s a set that contains nothing. But in exam problems, the trick is to check whether the description of the set actually allows any object to exist. You don’t guess — you test each condition logically.

Let’s go through each option one by one.

1. Option (i): Set of odd natural numbers divisible by 2

A natural number that is odd cannot be divisible by 2 — divisibility by 2 is the very definition of being even. So the condition “odd and divisible by 2” is impossible. No natural number satisfies both.

Therefore, this set has no elements — it is a null set.

2. Option (ii): Set of even prime numbers

A prime number has exactly two distinct positive divisors: 1 and itself. The number 2 is prime, and it is also even (the only even prime). So 2 belongs to this set.

Since there is at least one element, this set is not a null set.

3. Option (iii): $\{ x : x \text{ is a natural number}, x < 5 \text{ and } x > 7 \}$

The condition says $x < 5$ and $x > 7$ simultaneously. No real number can be both less than 5 and greater than 7 — the two inequalities have no overlap. Since natural numbers are a subset of real numbers, no natural number satisfies both.

Hence, this set is a null set.

4. Option (iv): $\{ y : y \text{ is a point common to any two parallel lines} \}$

Parallel lines, by definition, never intersect. They have no common point. So the set of points common to two parallel lines is empty.

Thus, this is a null set.

Method: Definition-Based Null Set Verification

Concept First

A null set (or empty set) is a set that contains no elements. To identify one, we check whether any object can satisfy the given condition. If none can, it’s a null set.

Steps

1. Read the condition carefully — understand what elements are being described.
2. Check if at least one element exists that satisfies all parts of the condition.
3. If no element exists → it is a null set.
4. If even one element exists → it is not a null set.

Applying to Each Option

(i) Set of odd natural numbers divisible by 2

• A number divisible by 2 is even.
• An odd number cannot be even.
• No number satisfies both → Null set ✓

(ii) Set of even prime numbers

• The only even prime number is 2.
• One element exists → Not a null set ✗

(iii) $\{ x : x \text{ is a natural number}, x < 5 \text{ and } x > 7 \}$

• No natural number can be both less than 5 and greater than 7.
• No element exists → Null set ✓

(iv) $\{ y : y \text{ is a point common to any two parallel lines} \}$

• Parallel lines never intersect — they have no common point.
• No point exists → Null set ✓

Final Answer

Null sets: (i), (iii), (iv)

Not a null set: (ii)

Here are the common mistakes students make when identifying null sets, using your question as the reference, along with how to avoid each.

Mistake 1: Misreading "Divisible by 2" as "Divisible by 1"

The error:

For set (i) — Set of odd natural numbers divisible by 2 — students think "odd numbers like 1, 3, 5... can be divided by 2 if we allow fractions." They forget that divisible in mathematics means divides exactly with remainder zero.

How to avoid:

• Remember: A number $n$ is divisible by 2 only if $n \div 2$ gives an integer.
• Odd numbers always leave remainder 1 when divided by 2.
• So no odd number is divisible by 2 → this set is empty (null set).

✓ Key rule: "Divisible by $k$" means $n/k$ is an integer, not just "can be divided."

Mistake 2: Forgetting that 2 is an Even Prime

The error:

For set (ii) — Set of even prime numbers — students say "there are no even primes" because they think all primes are odd. They forget the only exception: 2.

How to avoid:

• Memorise: 2 is the only even prime number.
• So the set $\{2\}$ is not empty — it has one element.
• Always check the definition of prime: a number greater than 1 with exactly two factors. 2 fits.

✓ Key rule: "Even prime" is not a contradiction — it's just the number 2.

Mistake 3: Ignoring the "and" Condition in Set-Builder Form

The error:

For set (iii) — $\{ x : x \text{ is a natural number}, x < 5 \text{ and } x > 7 \}$ — students pick numbers that satisfy one condition, like $x = 3$ (which is $<5$) or $x = 8$ (which is $>7$). They forget both must be true at the same time.

How to avoid:

• Read the condition as: "Find numbers that are simultaneously less than 5 and greater than 7."
• No natural number can be both $<5$ and $>7$ at once.
• So the set is empty (null set).

✓ Key rule: In set-builder notation, all conditions are joined by "and" — they must all hold together.

Mistake 4: Thinking Parallel Lines Meet at Infinity

The error:

For set (iv) — $\{ y : y \text{ is a point common to any two parallel lines} \}$ — students say "parallel lines meet at infinity, so there is a point." This is a misunderstanding of Euclidean geometry.

How to avoid:

• In standard Euclidean geometry (your syllabus), parallel lines never intersect.
• "At infinity" is a concept from projective geometry, not used in basic set theory or school geometry.
• So there is no common point → the set is empty (null set).

✓ Key rule: In school mathematics, parallel lines have no common point. Null set is correct.

Quick Summary Table

Final tip: Always write the definition of the set in your own words before deciding. If no object satisfies all conditions together, it's a null set.