List the elements in each set. See Example 1. {z|z is an integer less than or equal to 4}
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
0. Review of College Algebra
Functions
Problem 21
Textbook Question
List the elements in each set. See Example 1. {x|x is an irrational number that is also rational}
Verified step by step guidance1
Understand the definitions: A rational number is any number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). An irrational number is a number that cannot be expressed as such a fraction.
Analyze the set description: The set is defined as \(\{x \mid x \text{ is an irrational number that is also rational}\}\). This means we are looking for numbers that are both irrational and rational at the same time.
Recognize the logical contradiction: Since a number cannot be both rational and irrational simultaneously, there are no numbers that satisfy this condition.
Conclude the set elements: Because no number can be both irrational and rational, the set is empty.
Express the final answer: The set can be written as \(\emptyset\) or \(\{\}\), indicating it contains no elements.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Examples include 1/2, -3, and 0.75. They have either terminating or repeating decimal expansions.
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Rationalizing Denominators
Irrational Numbers
Irrational numbers cannot be expressed as a fraction of two integers. Their decimal expansions are non-terminating and non-repeating. Examples include √2, π, and e. They are distinct from rational numbers.
Recommended video:
Introduction to Complex Numbers
Set Definition and Intersection
A set is a collection of elements defined by a property. The question asks for elements that are both irrational and rational, which involves understanding the intersection of these sets. Since no number can be both rational and irrational, the intersection is the empty set.
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Parameterizing Equations Example 1
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