To answer whether it is correct to write [tex]\( A = \{m, a, t, h, e, m, a, t, i, c, s\} \)[/tex] to express the set of letters used in the word "mathematics," let's review the properties of mathematical sets and how they apply to this scenario:
1. Definition of a Set:
- A set is a collection of distinct elements without any specific order.
- Sets do not include duplicate elements.
2. Analysis of the Given Problem:
- The word "mathematics" contains the letters: m, a, t, h, e, m, a, t, i, c, s.
- As per the properties of sets, we need to consider each letter only once, even if it appears multiple times in the word.
3. Removing Duplicates:
- Identify each unique letter in "mathematics": m, a, t, h, e, i, c, s.
- These letters without duplication form the correct set of characters used in "mathematics".
4. Creating the Set:
- The unique letters identified form the set: [tex]\( A = \{m, a, t, h, e, i, c, s\} \)[/tex].
5. Verification:
- Presenting the elements, we list each letter only once: {m, a, t, h, e, i, c, s}.
- This satisfies the property of a set that contains distinct elements.
6. Correct Expression:
- Therefore, the correct way to express the set of letters used in "mathematics" is [tex]\( A = \{m, a, t, h, e, i, c, s\} \)[/tex] where there are no repeated elements.
Thus, it is not correct to write [tex]\( A = \{m, a, t, h, e, m, a, t, i, c, s\} \)[/tex] since sets must not contain duplicate elements. The correct representation of the set is [tex]\( A = \{m, a, t, h, e, i, c, s\} \)[/tex].
