Question - Solving Set Theory Problems with Roster Notation
Solution:
\text{There are five problems to solve in this question.}
\text{1. Convert the set-builder notation } \{x | x \text{ is a natural number less than 5}\} \text{ to roster form. }
\text{Solution: } \{1, 2, 3, 4\}
\text{2. Express the set } G = \{x | x \text{ is an even number between 1 and 20}\} \text{ in roster form. }
\text{Solution: } \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}
\text{3. Express the set } L = \{x | x \text{ is a letter in the word "MATHEMATICS"}\} \text{ in roster form. }
\text{Solution: } \{'M', 'A', 'T', 'H', 'E', 'I', 'C', 'S'\}
\text{4. Convert the set-builder notation } \{x | x \text{ is an odd integer greater than 5 and less than 15}\} \text{ to roster form. }
\text{Solution: } \{7, 9, 11, 13\}
\text{5. If } A = \{x | x \text{ is a prime number less than 10}\}, \text{ what is } A?
\text{Solution: } \{2, 3, 5, 7\}
Tutor Solution:
The set \( F \) in roster form is: \( F = \{1, 2, 3, 4\} \).
2. Express the set \( G = \{x \mid x \text{ is an even number between 1 and 20} \} \) in roster form. For this question, we need to list all even numbers between 1 and 20. Even numbers are multiples of 2, so we start with the first even number greater than 1, which is 2, and continue until we reach the last even number that is less than or equal to 20, which is 20.The set \( G \) in roster form is: \( G = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \).
3. Express the set \( L = \{x \mid x \text{ is a letter in the word "MATHEMATICS"} \} \) in roster form. Here we need to list each distinct letter that appears in the word "MATHEMATICS". Duplicate letters are not listed more than once in a set.The set \( L \) in roster form is: \( L = \{M, A, T, H, E, I, C, S\} \).
Note: We are not differentiating between upper and lower case since no such distinction is suggested. 4. Convert the set-builder notation \( J = \{x \mid x \text{ is an odd integer greater than 5 and less than 15} \} \) to roster form. In this question, we are looking for odd integers that are greater than 5 and less than 15. We start with the smallest odd integer greater than 5, which is 7, and continue up to the greatest odd integer less than 15, which is 13.The set \( J \) in roster form is: \( J = \{7, 9, 11, 13\} \).
5. If \( A = \{x \mid x \text{ is a prime number less than 10} \}, what is \( A \)? To solve this, we list all the prime numbers less than 10. Prime numbers are natural numbers greater than 1 that are not divisible by any other number except 1 and themselves. The prime numbers less than 10 are 2, 3, 5, and 7.The set \( A \) is: \( A = \{2, 3, 5, 7\} \).
And that's all the questions solved! Remember that set-builder notation is a compact way to define the criteria for membership in a set, while roster form lists out all elements in the set explicitly.CamTutor
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