Math Problem Statement
Solution
Let's analyze the problem step by step:
Problem Breakdown:
The set is defined as: This means contains all elements such that:
- is a rational number ().
- .
Analysis:
-
Solving : The solutions to this equation are:
-
Check if is a rational number:
- By definition, is an irrational number because it cannot be expressed as a ratio of two integers.
- Since , there are no rational solutions to .
Conclusion:
The set is empty (). Hence, the number of elements in , denoted as , is:
The answer should be re-evaluated for the options given. Let me know if you have additional clarifications. Here are related questions to explore:
Related Questions:
- What is the difference between rational and irrational numbers?
- Can ever have rational solutions for non-perfect square ?
- What is the formal proof that is irrational?
- How would this question change if included irrational numbers?
- What is the cardinality of a set and how is it calculated?
Tip:
Always verify whether the set constraints align with the number type specified (rational, integer, real, etc.).
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Math Problem Analysis
Mathematical Concepts
Sets
Rational and Irrational Numbers
Quadratic Equations
Formulas
x^2 = k (solutions for quadratic equations)
Theorems
Rational Root Theorem
Definition of Rational and Irrational Numbers
Suitable Grade Level
Grades 9-10
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