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Chapter 1: Problem 6

Let \(A=\\{x: x \in R,|x|<1\\}\) \(B=\\{x: x \in R,|x-1| \geq 1\\}\) and \(A \cup B=R-D\), then the set \(D\) is (A) \(\\{x: 1

Short Answer

Expert verified
The set \(D\) is \([1, 2)\), which matches option (B).

Step by step solution

01

Understand the Given Sets

First, understand the sets \(A\) and \(B\). Set \(A=\{x: x \in \mathbb{R},|x|<1\}\) includes all real numbers \(x\) such that \(-1<x<1\). Set \(B=\{x: x \in \mathbb{R},|x-1| \geq 1\}\) means \(|x-1|\) is greater than or equal to 1, which results in \(x \leq 0\) or \(x \geq 2\).
02

Union of Sets A and B

Next, find \(A \cup B\). Since \(A\) includes all \(x\) where \(-1<x<1\), and \(B\) includes \(x \leq 0\) or \(x \geq 2\), the union \(A \cup B\) results in all \(x\) such that \(-1<x<1\) or \(x \geq 2\). In interval notation, this is \((-1,1) \cup [2, \infty)\).
03

Find Missing Elements in R

To find set \(D\), note that \(A \cup B = \mathbb{R} - D\). Hence, \(D\) represents the elements not included in \((-1,1) \cup [2, \infty)\). This means \(D = [1, 2)\), as this interval is not covered by the union of \(A\) and \(B\).
04

Match with Options

Finally, match the set \(D = [1, 2)\) with the given options. The correct option is (B) \(\{x: 1 \leq x < 2\}\).

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Key Concepts

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

Real Numbers
Real numbers are a fundamental part of mathematics and include all the numbers that can be found on the number line. This set encompasses:
  • Natural numbers: 1, 2, 3, ...
  • Whole numbers: 0, 1, 2, 3, ...
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: fractions like 1/2, 3/4, including numbers that can be expressed as a ratio of two integers.
  • Irrational numbers: numbers like \sqrt{2} and \pi that cannot be expressed as a simple fraction.
Real numbers are denoted by \( \mathbb{R} \) and they make calculations, graphing, and analysis comprehensive and meaningful. To visualize real numbers, think of a number line stretching infinitely in both directions, where every point on this line corresponds to a real number.
Interval Notation
Interval notation is a concise way to describe subsets of real numbers. It helps in expressing ranges of numbers without listing every individual element. Here鈥檚 how it works:
  • Closed interval [a, b]: Includes all numbers between a and b, as well as a and b themselves. The square brackets denote this inclusion.
  • Open interval (a, b): Includes all numbers between a and b, but not a and b themselves. Round parentheses indicate exclusion.
  • Semi-open intervals [a, b) or (a, b]: One boundary is included and the other is not, as shown by a combination of square bracket and round parenthesis.
For instance, \( (-1, 1) \) represents all real numbers greater than -1 and less than 1. In contrast, \( [2, \infty) \) includes all real numbers from 2 up to infinity, including 2. This notation helps in analyzing and solving problems involving continuous data or sequences.
Union of Sets
The union of sets in set theory is an operation that combines elements from two or more sets into a single set. The basic idea is that if an element is present in any of the original sets, it will be in the union:
  • For example, if set \( A = \{1, 2, 3\} \) and set \( B = \{3, 4, 5\} \), then the union \( A \cup B = \{1, 2, 3, 4, 5\} \).
  • The symbol "\cup" is used to denote union.
  • Union is a commutative operation, meaning \( A \cup B = B \cup A \).
  • It is also associative, so \( (A \cup B) \cup C = A \cup (B \cup C) \).
In interval notation, if we have intervals \( (-1, 1) \) and \( [2, \infty) \), the union would be written as \( (-1, 1) \cup [2, \infty) \), representing all numbers in these intervals. The concept of union finds applications in probability, analysis, and algebra, making it indispensable in set operations.

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