document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166868>Question 1166868</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the definition of ⋆ does give a binary operation on the set. In case that ⋆ is not a binary operation, state the property it fails to satisfy and give a counterexample.\r<BR>\n" );
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document.write( "a). Let S = ℕ, define ⋆ by a ⋆ b=c, where in c is the largest integer less than the product of a and b. </SPAN>\n" );
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document.write( "Response revised/corrected....<br><BR>\n" );
document.write( "The two inputs are integers; and the output is always an integer.<br><BR>\n" );
document.write( "That makes * a binary operation on the set of integers.<br><BR>\n" );
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document.write( "Revision...<br><BR>\n" );
document.write( "I wrongly took the given set to be the set of positive integers; in standard nomenclature it is the set of natural numbers.<br><BR>\n" );
document.write( "Even if the set were the set of positive integers, there would be a single counterexample making * NOT a binary operation on the set:<BR>\n" );
document.write( "1*1 = 1-1 = 0<BR>\n" );
document.write( "But 0 is not a positive integer.<br><BR>\n" );
document.write( "With the set actually being the natural numbers, there are an infinite number of counterexamples.  If either a or b is 0, then a*b = 0-1 = -1; and -1 is not a natural number.<br><BR>\n" );
document.write( "ANSWER: a*b does NOT define a binary operation on the given set.<br>\r<BR>\n" );
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