Q.State whether each of the following statement is true or false. Justify your answer.

Concept: Two sets are disjoint if their intersection is empty, i.e., they share no common elements.

We check each pair by looking for any element that appears in both sets.

(i) $\{2, 3, 4, 5\}$ and $\{3, 6\}$

The element $3$ belongs to both sets, so their intersection is $\{3\} \neq \emptyset$.

False — the sets are not disjoint.

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

The element $a$ is common to both sets.

False — the sets are not disjoint.

(iii) $\{2, 6, 10, 14\}$ and $\{3, 7, 11, 15\}$

No element from the first set appears in the second (all even vs. all odd in this case). …

Understanding Disjoint Sets

Two sets are called disjoint when their intersection is empty — that is, they have absolutely no elements in common. Think of it as two groups that don't overlap at all.

The test is simple: scan through the elements of both sets. If you find even one element that belongs to both, the sets are not disjoint. If no such element exists, they are disjoint.

Mathematically, sets $A$ and $B$ are disjoint if and only if $A \cap B = \emptyset$.

Let me examine each pair:

(i) $\{2, 3, 4, 5\}$ and $\{3, 6\}$

1. List the elements of the first set: $2, 3, 4, 5$
2. List the elements of the second set: $3, 6$
3. Look for common elements: The number $3$ appears in both sets.
4. Since there is at least one common element, the intersection is $\{3\} \neq \emptyset$.

Statement (i) is FALSE. The sets are not disjoint because they share the element $3$.

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

1. First set contains the vowels: $a, e, i, o, u$
2. Second set contains: $a, b, c, d$
3. The letter $a$ appears in both sets.
4. Therefore $\{a, e, i, o, u\} \cap \{a, b, c, d\} = \{a\} \neq \emptyset$.

Statement (ii) is FALSE. These sets share the common element $a$.

(iii) $\{2, 6, 10, 14\}$ and $\{3, 7, 11, 15\}$

1. First set: $2, 6, 10, 14$ (all even numbers, specifically of the form $4n + 2$ for $n = 0, 1, 2, 3$)
2. Second set: $3, 7, 11, 15$ (all odd numbers, specifically of the form $4n + 3$ for $n = 0, 1, 2, 3$)
3. Check each element of the first set against the second:
   • Is $2$ in the second set? No.
   • Is $6$ in the second set? No.
   • Is $10$ in the second set? No.
   • Is $14$ in the second set? No. …

Here’s a clear, step-by-step solution method for checking whether two sets are disjoint.

Method: Common Element Check

Concept: Two sets are disjoint if they have no element in common.

That means their intersection is the empty set:

Steps

1. List the elements of both sets clearly.
2. Compare each element of the first set with every element of the second set.
3. If any element appears in both sets, the sets are not disjoint (false statement).
4. If no element is common, the sets are disjoint (true statement).

Applying the method

(i) $\{2, 3, 4, 5\}$ and $\{3, 6\}$

• Common element: 3 appears in both.
• False — they are not disjoint.

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

• Common element: a appears in both.
• False — they are not disjoint. …

🧠 Core Concept: Disjoint Sets

Two sets are disjoint if they have no element in common.

That means:

✗ Mistake #1: Confusing “disjoint” with “different”

Example of the mistake:

A student says (i) is false because “both sets have numbers, so they are not disjoint.”

Why it’s wrong:

Having the same type of elements (both are sets of numbers) does not make them non-disjoint. Only common elements matter.

✓ How to avoid:

Check element-by-element. For (i):

• Set 1: $\{2, 3, 4, 5\}$
• Set 2: $\{3, 6\}$
• Common element: 3 So they are not disjoint.

Answer (i): False — because $3$ belongs to both sets.

✗ Mistake #2: Forgetting to check all elements

Example of the mistake:

For (ii), a student sees $\{a, e, i, o, u\}$ and $\{a, b, c, d\}$ and says “they are different, so disjoint.”

Why it’s wrong:

They share the element a.

✓ How to avoid:

Scan every element of the first set against the second. Circle matches.

Answer (ii): False — because $a$ is common.

✗ Mistake #3: Assuming “no match” without checking carefully

Example of the mistake:

For (iii), a student quickly says “all numbers are different, so disjoint” — correct here, but they might miss a match in other problems.

Why it’s risky:

Speed leads to oversight. Always list the intersection.

✓ How to avoid:

Write:

• Set 1: $\{2, 6, 10, 14\}$
• Set 2: $\{3, 7, 11, 15\}$
• Intersection: $\emptyset$ → disjoint

Answer (iii): True — no common element.

✗ Mistake #4: Thinking “disjoint” means “no relationship at all”

Example of the mistake:

For (iv), a student says “both are sets of even and odd numbers, so they are related, hence not disjoint.”

Why it’s wrong:

Disjointness is about elements, not patterns or properties. …