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Chapter 14: Problem 42

Write the interval in sel-builder notation $$(-\infty, \infty)$$

Short Answer

Expert verified
In set-builder notation, the interval \(-\infty, \infty\) is written as \(x | x \in R\). This represents the set of all \(x\) that are real numbers.

Step by step solution

01

Understanding Interval Notation

The interval notation \(-\infty, \infty\) includes all real numbers. The parentheses indicate that the bounds, negative infinity and positive infinity, are not included in the set because they are not real numbers. Strictly speaking, infinity and negative infinity are not numbers, but rather concepts used to describe an unbounded quantity.
02

Translating to Set-Builder Notation

Set-builder notation is a method used to describe a set by stating the properties that its members must satisfy. Here, to describe the set of all real numbers, which is what the interval \(-\infty, \infty\) represents, you can use the set-builder notation as follows: \(x | x \in R\). This describes the set of all \(x\) such that \(x\) is a real number.

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Key Concepts

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

Interval Notation
Interval notation is a shorthand way to define sets of real numbers. It tells us what numbers are included or excluded in a range. When you see \((-\infty, \infty)\), it symbolizes the set of all real numbers. Parentheses \(()\) mean that the endpoints are not included. Here, the endpoints are negative and positive infinity, which are not real numbers. They symbolize an unlimited continuation in their respective directions.
  • \((-\infty, a)\) means all numbers less than \(a\).
  • \((b, \infty)\) means all numbers greater than \(b\).
  • \((a, b)\) includes all numbers between \(a\) and \(b\), excluding the endpoints.
  • \([a, b]\) includes \(a\) and \(b\).
Interval notation is an efficient way to express sets of numbers, especially when dealing with real number lines.
Real Numbers
Real numbers are all the numbers you can think of that are not imaginary. They include rational numbers like fractions and decimals, whole numbers, and irrational numbers like square roots of non-square numbers. These numbers can be put on a line called the number line.
  • Rational numbers: Examples are \(\frac{1}{2}\), \(-3\), and \(4.7\).
  • Irrational numbers: Examples are \(\sqrt{2}\) and \(\pi\).
Real numbers are everywhere in our everyday world, and they help us count, measure, and calculate many things.
Set Theory
Set Theory is a branch of mathematics that studies collections of objects, called sets. It helps us understand and organize different mathematical concepts.
  • In Set Theory, a set is a collection of distinct objects, considered as an object itself. An example is {1, 2, 3}, which is a set of numbers.
  • 'Set-Builder Notation' is a way to specify a set by describing the properties its members share. For example, \( \{ x \mid x > 1 \} \) means all real numbers greater than one.
  • 'Union' and 'Intersection' are operations that combine sets or find common elements. For instance, the union of sets A and B \( (A \cup B) \) includes all elements in either set A or B.
Understanding Set Theory helps us work with complex mathematical problems by breaking them into understandable pieces.

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Most popular questions from this chapter

Graph the set. (GRAPH CANNOT COPY) $$\\{x | 0 \leq x<4\\}$$

Solve and write the answer in interval notation. $$2 x+4

Comparing Services A site licensing fee for a computer program is \(\$ 1500\). Paying this fee allows the company to use the program at any computer terminal within the company. Alternatively, the company can choose to pay \(\$ 200\) for each individual computer it owns. How many individual computers must a company own for the site license to be the more economical choice for the company?

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