State whether the following sets are empty finite or infinite sets., mention the cardinal number.
{x|x is a vowel in the word MATHEMATICS}

Answer

To determine whether the set {x | x is a vowel in the word MATHEMATICS} is empty, finite, or infinite, and to mention the cardinal number, we can follow these steps:

### Step 1: Identify the vowels in the word "MATHEMATICS"
- The vowels in the English alphabet are A, E, I, O, and U.
- Now, let's look at the letters in the word "MATHEMATICS": M, A, T, H, E, M, A, T, I, C, S.

### Step 2: List the vowels present in "MATHEMATICS"
- From the letters above, we can identify the vowels:
- A (appears twice)
- E (appears once)
- I (appears once)

### Step 3: Create the set of unique vowels
- The unique vowels we found are: {A, E, I}.
- Note that we only count each vowel once, regardless of how many times it appears in the word.

### Step 4: Determine the nature of the set
- Since we have identified three unique vowels, the set is not empty.
- The set contains a finite number of elements (3 in this case).

### Step 5: State the cardinal number
- The cardinal number of the set, which represents the number of elements in the set, is 3.

### Conclusion
- The set {x | x is a vowel in the word MATHEMATICS} is a finite set, and the cardinal number is 3.

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