Find whether the sets A and B are disjoint or overlapping sets :
(i) A = { x | x is a letter in the word MODERN } and
B = { x | x is a letter in the word MAYOR }

Answer

To determine whether the sets A and B are disjoint or overlapping, we will follow these steps:

### Step 1: Identify the elements of Set A
Set A consists of the letters in the word "MODERN".
- The unique letters in "MODERN" are:
- M, O, D, E, R, N

So, we can write:
\[ A = \{ M, O, D, E, R, N \} \]

### Step 2: Identify the elements of Set B
Set B consists of the letters in the word "MAYOR".
- The unique letters in "MAYOR" are:
- M, A, Y, O, R

So, we can write:
\[ B = \{ M, A, Y, O, R \} \]

### Step 3: Find the intersection of Set A and Set B
The intersection of two sets, denoted as \( A \cap B \), is the set of elements that are common to both sets.
- From Set A: M, O, D, E, R, N
- From Set B: M, A, Y, O, R

Now, we identify the common elements:
- Common elements are M and O.

Thus, we have:
\[ A \cap B = \{ M, O \} \]

### Step 4: Determine if the sets are disjoint or overlapping
- Sets are disjoint if they have no elements in common (i.e., \( A \cap B = \emptyset \)).
- Sets are overlapping if they have at least one element in common (i.e., \( A \cap B \neq \emptyset \)).

Since \( A \cap B = \{ M, O \} \) contains elements, we conclude that:
- Sets A and B are overlapping sets.

### Final Answer:
Sets A and B are overlapping sets.

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