The set of real numbers between -1 and 0 is infinite, finite, 0, or 10?
Understand the Problem
The question is asking to classify the set of real numbers that lie between -1 and 0. This involves understanding concepts of sets and infinity in mathematics.
Answer
The set of real numbers between -1 and 0 is $S = \{ x \in \mathbb{R} : -1 < x < 0 \}$.
Answer for screen readers
The set of real numbers that lie between -1 and 0 is: $$ S = { x \in \mathbb{R} : -1 < x < 0 } $$
Steps to Solve
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Identify the interval The interval we need to classify is between -1 and 0. We denote this interval in interval notation as $(-1, 0)$, which means all real numbers greater than -1 and less than 0.
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Determine the type of set Since the interval $(-1, 0)$ includes all real numbers between -1 and 0, it is an open interval. This means the endpoints -1 and 0 are not included in the set.
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Visualize the interval It's helpful to visualize this on a number line where you would place an open circle at -1 and an open circle at 0, indicating those points are not part of the set but all numbers in between are included.
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Express as a set We can express the set of real numbers that lie between -1 and 0 as: $$ S = { x \in \mathbb{R} : -1 < x < 0 } $$ This notation means that $S$ includes all real numbers $x$ such that $x$ is greater than -1 and less than 0.
The set of real numbers that lie between -1 and 0 is: $$ S = { x \in \mathbb{R} : -1 < x < 0 } $$
More Information
The interval $(-1, 0)$ represents all decimal numbers, fractions, and irrational numbers that exist between these two endpoints, such as -0.5, -0.1, or -0.9. Understanding intervals is a key element in set theory and helps in solving a variety of problems in mathematics.
Tips
- Confusing open intervals with closed intervals: Remember, open intervals do not include the endpoints, while closed intervals do.
- Misinterpretation of the interval notation. Ensure that $(-1, 0)$ means all numbers greater than -1 and less than 0, not including -1 and 0 themselves.
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