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Chapter 1: Problem 20

Fill in the blanks. The set \(\\{x | x \text { is a whole number. }\\}\) is read as "______" .

Short Answer

Expert verified
The set of all x such that x is a whole number.

Step by step solution

01

Interpret the Set Notation

Look at the set notation \(\{x | x \text{ is a whole number.}\}\). Here, 'x' is an arbitrary member of the set. The condition for 'x' to be in this set, as stated after '|', is that 'x' is a whole number.
02

Convert Mathematical Notation into Sentence

Using this understanding, convert the set notation into a sentence. The set can read as: The set of all x such that x is a whole number.

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Key Concepts

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

Whole Numbers
Whole numbers are simple yet fundamental mathematical concepts. They include all the non-negative numbers beginning from zero, followed by one, two, three, and so on. Unlike integers, which can be negative or positive, whole numbers are never negative.

Here's what you need to know about whole numbers:
  • They start at zero and go up infinitely: 0, 1, 2, 3, 4, ...
  • They are used in counting and ordering.
  • Whole numbers do not include fractions or decimals.
  • While zero is a whole number, it is neither positive nor negative.
When identifying whole numbers in a mathematical context, ensure there's always an absence of decimals or negative signs. This makes whole numbers useful in scenarios where only complete units are considered, like counting physical items.
Mathematical Notation
Mathematical notation is a symbolic language that allows mathematicians and students to express mathematical ideas succinctly and precisely. In the exercise, the set notation uses this language to describe a group of numbers with specific properties.

Some key features of mathematical notation are:
  • Symbols like \( \{ \} \) are used to denote sets. These curly braces indicate that we're talking about a set of elements.
  • The vertical bar \( | \) within a set notation acts as a separator, meaning 'such that'.
  • Variables such as \( x \) are employed to represent members of the set.
  • Relational expressions after the vertical bar (\( | \)) serve as conditions for membership in the set.
Understanding and using mathematical notation is crucial as it provides a universal and simplified way of conveying complex mathematical concepts. Distinguishing between symbols and their meanings is key to interpreting problems and equations correctly.
Interpreting Sets
Interpreting sets involves breaking down symbolic notation into understandable sentences. This is crucial for developing a deeper understanding of mathematical concepts beyond just recognizing symbols.

Let's consider the process to convert set notation into a statement:
  • Begin by reading the symbol inside the curly braces. For example, \( \{x \} \) indicates we are discussing a set containing elements called \( x \).
  • The vertical bar \( | \) translates to 'such that'. This is your cue to look for additional conditions or properties of the set.
  • Everything after the bar describes the rule or conditions for \( x \). In this exercise, it's noted that \( x \) is a whole number.
  • Combine these interpretations into a comprehensive sentence, for instance, "The set of all \( x \) such that \( x \) is a whole number."
By interpreting sets this way, you transform the abstract into something tangible, making it easier to tackle complex mathematical problems.

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Most popular questions from this chapter

Let \(x=8, y=4,\) and \(z=2 .\) Write each phrase as an algebraic expression, and evaluate it. Assume that no denominators are \(0 .\) \(z\) less than \(y\)

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Use the given property to rewrite the expression in a different form. SEE EXAMPLE \(6 .\) (OBJECTIVE 5 ) \((x y) z ;\) associative property of multiplication.

Which property of real numbers justifies each statement? $$9 \cdot \frac{1}{9}=1$$

What is the purpose of using variables? Why aren't ordinary numbers enough?
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