List the elements in each set. See Example 1. {x|x is an irrational number that is also rational}

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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.

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.

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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