text: # 域中分數場
  
  ## 等價設定
  
  ${\displaystyle {\textbf {D}}}$ 係一個域。將${\displaystyle P={\textbf {D}}\backslash ({\textbf {D}}^{\times }\cup \{0\})=\{(a,b):a,b\in {\textbf {D}},b\neq 0\}}$
  
  • 自反性${\displaystyle (a,b)\equiv (a,b)\iff ab=ab}$
  • 對稱性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb\iff (c,d)\equiv (a,b)}$
  • 傳遞性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb}$ 同時${\displaystyle (c,d)\equiv (e,f)\iff cf=de}$ ${\displaystyle af=be\iff (a,b)\equiv (e,f)}$
  
  ${\displaystyle P/\equiv }$ 叫做${\displaystyle Q({\textbf {D}})}$ ，而${\displaystyle Q({\textbf {D}})}$ 就係分數場（Field of Fractions）
  
  ## 加法同乘法
  
  ${\displaystyle Q({\textbf {D}})}$ 入面，都會有加法同乘法，而佢哋都係普通嘅加同乘。
  
  ${\displaystyle {\frac {a}{b}}={\frac {a'}{b'}}}$ ${\displaystyle {\frac {c}{d}}={\frac {c'}{d'}}}$
  
  ${\displaystyle ab'=a'b}$ ${\displaystyle cd'=c'd}$
  
  ${\displaystyle (ad+bc)b'd'=ab'd'+bb'cd'=a'bdd'+bb'c'd=(a'd'+b'c')bd}$
  
  ${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a'}{b'}}\cdot {\frac {c'}{d'}}}$
explanation: The text extract is a mathematical exposition on the concept of a field of fractions (分數場) in abstract algebra. It defines the set \( P \) and the equivalence relation \( \equiv \) on \( P \), and then describes the properties of this relation (reflexivity, symmetry, and transitivity). It also introduces the field of fractions \( Q({\textbf{D}}) \) and explains how addition and multiplication are defined in this field.
  
  The document does not contain a specific question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 域中分數場
  
  ## 等價設定
  
  ${\displaystyle {\textbf {D}}}$ 係一個域。將${\displaystyle P={\textbf {D}}\backslash ({\textbf {D}}^{\times }\cup \{0\})=\{(a,b):a,b\in {\textbf {D}},b\neq 0\}}$
  
  • 自反性${\displaystyle (a,b)\equiv (a,b)\iff ab=ab}$
  • 對稱性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb\iff (c,d)\equiv (a,b)}$
  • 傳遞性${\displaystyle (a,b)\equiv (c,d)\iff ad=cb}$ 同時${\displaystyle (c,d)\equiv (e,f)\iff cf=de}$ ${\displaystyle af=be\iff (a,b)\equiv (e,f)}$
  
  ${\displaystyle P/\equiv }$ 叫做${\displaystyle Q({\textbf {D}})}$ ，而${\displaystyle Q({\textbf {D}})}$ 就係分數場（Field of Fractions）
  
  ## 加法同乘法
  
  ${\displaystyle Q({\textbf {D}})}$ 入面，都會有加法同乘法，而佢哋都係普通嘅加同乘。
  
  ${\displaystyle {\frac {a}{b}}={\frac {a'}{b'}}}$ ${\displaystyle {\frac {c}{d}}={\frac {c'}{d'}}}$
  
  ${\displaystyle ab'=a'b}$ ${\displaystyle cd'=c'd}$
  
  ${\displaystyle (ad+bc)b'd'=ab'd'+bb'cd'=a'bdd'+bb'c'd=(a'd'+b'c')bd}$
  
  ${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a'}{b'}}\cdot {\frac {c'}{d'}}}$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a mathematical exposition on the concept of a field of fractions (分數場) in abstract algebra. It defines the set \( P \) and the equivalence relation \( \equiv \) on \( P \), and then describes the properties of this relation (reflexivity, symmetry, and transitivity). It also introduces the field of fractions \( Q({\textbf{D}}) \) and explains how addition and multiplication are defined in this field.
  
  The document does not contain a specific question or a question and answer.
  
  Label: no-question