document.write( "Question 1184157: Construct two examples of bijective function from ZxZ to Q where Z is set of all integers and Q is set of rational numbers? \n" ); document.write( "
Algebra.Com's Answer #849929 by ikleyn(52782)\"\" \"About 
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\n" ); document.write( "Construct two examples of bijective function from ZxZ to Q where Z is set
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\n" ); document.write( "\n" ); document.write( "        In his post,  @CPhill states and insists that there is no bijective function from  Z×Z  to  Q. \r
\n" ); document.write( "\n" ); document.write( "        \"The difference in their  \" denseness \"  properties prevents such a mapping from being constructed.\"\r
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\n" ); document.write( "\n" ); document.write( "        It is not correct,  and below I explain  \"highlight%28highlight%28WHY%29%29\".\r
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document.write( "The set ZxZ is countably infinite.  It means that there is a bijection \r\n" );
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document.write( "    f: ZxZ  <---> Z\r\n" );
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document.write( "The set Q is also countably infinite. It means that there is a bijection\r\n" );
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document.write( "    g: Z <---> Q\r\n" );
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document.write( "It  \"highlight%28highlight%28IMPLIES%29%29\"  that the composition  gof  is the bijection ZxZ and Q\r\n" );
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document.write( "            f             g\r\n" );
document.write( "    ZxZ <--------> Z <--------> Q.\r\n" );
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document.write( "It disproves the statement by @CPhill.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Another question is IF it is possible to construct an explicit bijective function .\r
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\n" ); document.write( "\n" ); document.write( "I asked Google. Google AI created this response under this link\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=Construct+two+examples+of+bijective+function+from+ZxZ+to+Q+where+Z+is+set+of+all+integers+and+Q+is+set+of+rational+numbers%3F&rlz=1C1CHBF_enUS1071US1071&oq=Construct+two+examples+of+bijective+function+from+ZxZ+to+Q+where+Z+is+set+of+all+integers+and+Q+is+set+of+rational+numbers%3F&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIGCAEQRRhA0gEJMTg1NWowajE1qAIIsAIB8QXIg8hlL4FHy_EFyIPIZS-BR8s&sourceid=chrome&ie=UTF-8\r
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document.write( "             - - - START of the Google response - - - (of 2/21/2025)\r\n" );
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document.write( "Two examples of bijective functions from ℤxℤ to ℚ (where ℤ is the set of all integers and ℚ is the set \r\n" );
document.write( "of all rational numbers) are: \r\n" );
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document.write( "Example 1: Function using a numerator-denominator pairing \r\n" );
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document.write( "• Definition: f(m, n) = \r\n" );
document.write( "	• If n ≠ 0, then f(m, n) = m/n \r\n" );
document.write( "	• If n = 0, then f(m, n) = m \r\n" );
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document.write( "Explanation: \r\n" );
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document.write( "• This function maps each ordered pair (m, n) from ℤxℤ to a unique rational number. \r\n" );
document.write( "• When n is not zero, it represents the rational number \"m/n\". \r\n" );
document.write( "• When n is zero, the function effectively maps the pair to infinity (or any other fixed value depending on your interpretation of division by zero). \r\n" );
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document.write( "Example 2: Function using a \"diagonal\" mapping with a sign convention \r\n" );
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document.write( "• Definition: f(m, n) = (2m + n) / (2*(abs(n) + 1)) \r\n" );
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document.write( "Explanation: \r\n" );
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document.write( "• This function creates a unique mapping by leveraging the \"diagonal\" pattern of adding 2m and n, while ensuring that the denominator always has a positive even value. \r\n" );
document.write( "• The absolute value of n in the denominator is used to handle cases where n is negative, preventing duplicate values. \r\n" );
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document.write( "Key points about bijective functions: [1, 2]  \r\n" );
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document.write( "• A bijective function needs to be both injective (one-to-one) and surjective (onto), meaning each element in the domain maps to a unique element in the codomain, and every element in the codomain is reached by some element in the domain. [1, 2]  \r\n" );
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document.write( "Generative AI is experimental.\r\n" );
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document.write( "             - - - END of the Google response - - - \r\n" );
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\n" ); document.write( "\n" ); document.write( "                 Below is my comment about the Google AI response\r
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\n" ); document.write( "\n" ); document.write( "Example 1 above is  INCORRECT.  The function defined in  Example 1 is  NOT  bijective.\r
\n" ); document.write( "\n" ); document.write( "It is not bijective,  since the pairs  (m,1)  and  (m,0) map into the same rational/(integer)  number  m.\r
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\n" ); document.write( "\n" ); document.write( "                Surely,  I reported to  Google about it.\r
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\n" ); document.write( "\n" ); document.write( "Example 2  looks like to be correct.  At least,  I do not see the way to disprove it.\r
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