Q.Which of the following pairs of sets are disjoint

Concept: Disjoint Sets

Two sets are disjoint if their intersection is the null set, i.e., they have no common elements.

(i) $A = \{1, 2, 3, 4\}$ and $B = \{x : x \text{ is a natural number and } 4 \le x \le 6\} = \{4, 5, 6\}$

The element $4$ belongs to both sets, so $A \cap B = \{4\} \neq \emptyset$. Not disjoint.

(ii) $A = \{a, e, i, o, u\}$ and $B = \{c, d, e, f\}$

The element $e$ is common to both sets, so $A \cap B = \{e\} \neq \emptyset$. Not disjoint. …

The concept of disjoint sets is beautifully simple: two sets are disjoint if their intersection is empty, meaning they have absolutely nothing in common. Think of it as two circles that never touch. To determine whether sets are disjoint, we need to examine their elements carefully and ask: is there even one element that belongs to both?

Let me work through each pair systematically.

(i) $\{1, 2, 3, 4\}$ and $\{x : x \text{ is a natural number and } 4 \le x \le 6\}$

1. Write out the second set explicitly. The natural numbers satisfying $4 \le x \le 6$ are simply $4, 5, 6$. So the second set is $\{4, 5, 6\}$.

2. Look for common elements. The first set contains $1, 2, 3, 4$. The second contains $4, 5, 6$. The element $4$ appears in both sets.

3. Conclude. Since the intersection $\{1, 2, 3, 4\} \cap \{4, 5, 6\} = \{4\} \neq \emptyset$, these sets are not disjoint.

(ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$

1. Identify the sets. The first set contains the five vowels. The second set contains four consonants and one vowel.

2. Spot the overlap. The letter $e$ appears in both sets.

3. Conclude. The intersection is $\{e\} \neq \emptyset$, so these sets are not disjoint.

(iii) $\{x : x \text{ is an even integer}\}$ and $\{x : x \text{ is an odd integer}\}$ …

Method: Definition-Based Null Intersection Test

Two sets are disjoint if they have no elements in common — i.e., their intersection is the empty set ($\emptyset$).

Steps

1. Write both sets clearly — list elements or describe the pattern.
2. Find the intersection — identify common elements.
3. Check if intersection is empty:
   • If yes → sets are disjoint.
   • If no → sets are not disjoint.

(i) $\{1, 2, 3, 4\}$ and $\{x : x \text{ is a natural number and } 4 \leq x \leq 6\}$

• Set 1: $\{1, 2, 3, 4\}$
• Set 2: $\{4, 5, 6\}$ (natural numbers from 4 to 6)
• Intersection: $\{4\}$ — not empty
• Result: Not disjoint

(ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$

• Set 1: $\{a, e, i, o, u\}$
• Set 2: $\{c, d, e, f\}$
• Intersection: $\{e\}$ — not empty
• Result: Not disjoint

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Common Mistakes in Null Set / Disjoint Set Identification

Mistake 1: Misreading the Set-Builder Notation

The Error: Students often misread {x : x is a natural number and 4 ≤ x ≤ 6} as including 4, 5, 6 and all numbers from 4 to 6 in the first set. They forget that the first set {1, 2, 3, 4} already contains 4.

Why it happens: The notation 4 ≤ x ≤ 6 means x is greater than or equal to 4 AND less than or equal to 6. So the second set is {4, 5, 6}.

How to avoid: Always write out the elements of the set-builder form explicitly before comparing:

• Set A = {1, 2, 3, 4}
• Set B = {4, 5, 6}
• Common element = 4 → Not disjoint.

Mistake 2: Forgetting the Definition of "Disjoint"

The Error: Students think "disjoint" means "different" or "not equal." They see {a, e, i, o, u} and {c, d, e, f} and say "they're different, so they're disjoint."

Why it happens: The word "disjoint" sounds like "separate" in everyday language.

Correct Definition: Two sets are disjoint if they have no element in common — their intersection is the empty set:

How to avoid: Always check for common elements — not just whether the sets look different. In this pair, e appears in both → Not disjoint.

Mistake 3: Confusing "Even" and "Odd" with Specific Numbers

The Error: Students think {x : x is an even integer} and {x : x is an odd integer} share no numbers, but then doubt themselves thinking "what about 0?" or "what about negative numbers?"

Why it happens: They forget the definition: an even integer is any integer divisible by 2 (..., -4, -2, 0, 2, 4, ...). An odd integer is any integer not divisible by 2 (..., -3, -1, 1, 3, ...).

The Truth: These two sets are truly disjoint — no integer can be both even and odd.

How to avoid: Remember the fundamental property:

• Even numbers: $2k$ for some integer $k$
• Odd numbers: $2k+1$ for some integer $k$
• These forms can never be equal → Disjoint.

Mistake 4: Not Checking All Elements Carefully …