Math Problem Statement

If S = {x / x ∈ Q, x^2 = 5}, then n P(S) = ?

Solution

Let's analyze the problem step by step:

Problem Breakdown:

The set SS is defined as: S={xxQ,x2=5}S = \{ x \mid x \in \mathbb{Q}, x^2 = 5 \} This means SS contains all elements xx such that:

  1. xx is a rational number (xQx \in \mathbb{Q}).
  2. x2=5x^2 = 5.

Analysis:

  1. Solving x2=5x^2 = 5: The solutions to this equation are: x=±5.x = \pm \sqrt{5}.

  2. Check if 5\sqrt{5} is a rational number:

    • By definition, 5\sqrt{5} is an irrational number because it cannot be expressed as a ratio of two integers.
    • Since 5Q\sqrt{5} \notin \mathbb{Q}, there are no rational solutions to x2=5x^2 = 5.

Conclusion:

The set SS is empty (S=S = \emptyset). Hence, the number of elements in SS, denoted as n(P(S))n(P(S)), is: 0.\boxed{0}.

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:

  1. What is the difference between rational and irrational numbers?
  2. Can x2=kx^2 = k ever have rational solutions for non-perfect square kk?
  3. What is the formal proof that 5\sqrt{5} is irrational?
  4. How would this question change if SS included irrational numbers?
  5. 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