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Chapter 1: Problem 2

Write each interval in set notation and graph it on the real line. $$ (-3,5] $$

Short Answer

Expert verified
Set notation: \( \{ x \ | \ -3 < x \leq 5 \} \); graph: line between open circle at \(-3\) and closed circle at 5.

Step by step solution

01

Understand Interval Notation

The interval \((-3, 5]\) means that the set includes all numbers between -3 and 5. The parenthesis \((-3\) indicates that -3 is not included in the set, and the bracket \([5]\) indicates that 5 is included in the set.
02

Convert to Set Notation

In set notation, this interval is expressed as \ \{ x \ | \ -3 < x \leq 5 \}, \ which reads 'the set of all \(x\) such that \(-3\) is less than \(x\) and \(x\) is less than or equal to 5.'
03

Prepare to Graph the Interval

To graph \((-3, 5]\) on the real line, first identify the numbers \(-3\) and \(5\). Then decide how they will be represented: \(-3\) with an open circle (not included) and \(5\) with a closed circle (included).
04

Draw the Graph

On the number line, draw an open circle at \(-3\) indicating that -3 is not included, and a closed circle at 5 indicating that 5 is included. Draw a line connecting the points between \(-3\) and \(5\) to show all the numbers in between are included in the set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Line Graphing
Real line graphing is a powerful visual tool that helps us understand intervals and sets. On the real number line, numbers are usually represented as points located at regular intervals. In the case of open and closed intervals, the representation on the graph helps us determine the inclusivity of the boundary points.
For graphing intervals like (-3, 5], we:
  • First, mark the numbers at -3 and 5 on the line.
  • Using an open circle at -3, we show it鈥檚 not included in the interval.
  • Then, with a closed circle at 5, we indicate that it's included.
  • Finally, draw a line between -3 and 5 to demonstrate that all numbers in between are part of the interval.
Graphing on the real line provides a clear, visual representation of the range of numbers that belong to the interval.
Set Notation
Set notation is a concise way to describe a collection of numbers, providing a formal language for mathematics. It can clearly define which numbers are included in a set by means of specific conditions.
For the interval (-3, 5], the transformation into set notation would look like this:
  • It's written as \( \{ x \mid -3 < x \leq 5 \} \), which reads as "the set of all x such that -3 is less than x and x is less than or equal to 5."
  • This format clarifies the parameters of the interval by explicitly stating the conditions under which numbers belong to the set.
Set notation is efficient in communicating mathematical ideas when dealing with intervals and is often used in conjunction with other notations like interval notation and graph representations.
Open and Closed Intervals
Understanding open and closed intervals is key to mastering interval notation. It is crucial to distinguish between open, closed, and half-open (or half-closed) intervals because this impacts the set's boundaries.
  • **Open intervals** (*a, b*): Neither endpoint is included. Both parentheses, like *,*, indicate openness.
  • **Closed intervals** *[a, b]*: Both endpoints are included. The brackets, like *[*, *]*, show closure.
  • **Half-open intervals** (*a, b]* or *[a, b*): Only one endpoint is included, illustrated by mixing a parenthesis and a bracket.
For (-3, 5], we have a half-open interval:
  • The number -3 is not part of the set, indicated by the open parenthesis.
  • The number 5 is included in the set, marked by the closed bracket.
This differentiation is important for defining precise boundaries and behaviors of sets in mathematics.

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