Problem 3 - Precalculus Mathematics for Calculus

Question

The set of numbers between but not including 2 and 7 can be written as follows: ______ in set-builder notation and _____ in interval notation.

Step-by-Step Solution

Answer

Set-builder: \( \{ x \mid 2 < x < 7 \} \); Interval: \( (2, 7) \).

Step 1: Understanding the Problem

We need to express the set of numbers between 2 and 7 in two different notations: set-builder notation and interval notation. The numbers 2 and 7 are not included in the set, which means we need open intervals or exclusive conditions.

Step 2: Set-Builder Notation

Set-builder notation is a way of specifying a set by stating the properties that its members must satisfy. For a set of numbers greater than 2 and less than 7, we can express this as: \( \{ x \mid 2 < x < 7 \} \) This means 'the set of all x such that x is greater than 2 and less than 7.'

Step 3: Interval Notation

Interval notation is a way of writing subsets of the real numbers. For numbers strictly between 2 and 7, we use parentheses to denote that the endpoints are not included. The interval notation for this set is: \( (2, 7) \) This represents all numbers greater than 2 and less than 7.

Key Concepts

Set-Builder Notation

When talking about sets in mathematics, set-builder notation is a useful method for defining a set by describing the properties that its elements share. This notation efficiently expresses sets, especially when dealing with larger sets or those defined by similar characteristics.
Set-builder notation is usually written in the form:

• A pair of curly braces \( \{ \} \) to enclose the entire description.
• A variable, such as \( x \), that represents any element of the set.
• A vertical line \( \mid \), which can be read as "such that."
• A condition that describes the elements, like inequalities or equations.

For example, if you want to define the set of numbers greater than 2 and less than 7, you would write this in set-builder notation as:\[ \{ x \mid 2 < x < 7 \} \]
This means "the set of all \( x \) such that \( x \) is greater than 2 and less than 7." Because numbers 2 and 7 are not included, it perfectly illustrates the open nature of the set.

Interval Notation

Interval notation provides a concise way of describing intervals of numbers on the real number line. It is particularly beneficial because of its simplicity and ease of reading.
This notation uses parentheses and brackets to show whether endpoints are included or not:

• Parentheses \( ( ) \) are used for open intervals, where endpoints are excluded.
• Brackets \( [ ] \) indicate closed intervals, where endpoints are included.

For an open interval like the one between the numbers 2 and 7, you use parentheses:\( (2, 7) \)
Here, \( 2 \) and \( 7 \) are the boundaries of the interval, and neither are part of the set itself. This notation captures all numbers greater than 2 and less than 7, emphasizing the exclusion of the boundary numbers.

Open Intervals

Open intervals are an essential concept in understanding the set of real numbers as they signify certain ranges between numbers to be exclusive. They play a significant role in calculus and analysis, providing a groundwork in defining domains and ranges.
An open interval is denoted using parentheses. For instance, the interval between 2 and 7 is represented as \( (2, 7) \). This means all numbers that lie strictly between 2 and 7, not including 2 and 7 themselves.
Some key features of open intervals include:

• The endpoints are not part of the interval. This means, in the set \( (2, 7) \), only numbers like 3, 4, and 5, etc., but neither 2 nor 7.
• They are often used when defining domains where specific values need to be excluded, allowing flexibility in function evaluation.

Open intervals are crucial when working with functions that have undefined points at specific values, ensuring that calculations are managed smoothly.