Convert the following set-builder notation to roster notation: {x | x is a natural number and 23 < x < 27}. Convert the following roster notation to set-builder notation: {5, 6, 7,... Convert the following set-builder notation to roster notation: {x | x is a natural number and 23 < x < 27}. Convert the following roster notation to set-builder notation: {5, 6, 7, 8, ...}
Understand the Problem
The problem requires converting between set-builder notation and roster notation for sets of numbers. For part (a), you need to list all natural numbers between 23 and 27 in roster form. For part (b), given a roster notation {5, 6, 7, 8, ...}, you need to determine the correct set-builder notation.
Answer
(a) $\{24, 25, 26\}$ (b) $\{x \mid x \text{ is a natural number and } x \geq 5 \}$
Answer for screen readers
(a) ${24, 25, 26}$ (b) ${x \mid x \text{ is a natural number and } x \geq 5 }$
Steps to Solve
- Convert set-builder to roster notation for part (a)
The set-builder notation ${x \mid x \text{ is a natural number and } 23 < x < 27}$ describes the set of all natural numbers $x$ such that $x$ is greater than 23 and less than 27. The natural numbers that satisfy this condition are 24, 25, and 26.
Therefore the roster notation is ${24, 25, 26}$.
- Convert roster to set-builder notation for part (b)
The roster notation ${5, 6, 7, 8, ...}$ describes the set of all natural numbers starting from 5 and continuing indefinitely. This can be described in set-builder notation as the set of all $x$ such that $x$ is a natural number and $x$ is greater than or equal to 5. This can be written as ${x \mid x \text{ is a natural number and } x \geq 5 }$.
(a) ${24, 25, 26}$ (b) ${x \mid x \text{ is a natural number and } x \geq 5 }$
More Information
Set-builder notation provides a rule for determining membership in a set, while roster notation explicitly lists the elements of the set.
Tips
A common mistake in part (a) would be including 23 or 27 in the roster form, but the inequality is strict ($23 < x < 27$), so these values are not included. In part (b), a common mistake is to choose $x > 5$ instead of $x \geq 5$, but 5 appears in the roster form, so it should be included in the set.
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