document.write( "Question 1074715: Prove or disprove the statements below.
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\n" ); document.write( "\n" ); document.write( "I am not sure where to start, could someone please give me some hints on how to solve these kind of question. Thank you in advance. \n" ); document.write( "
Algebra.Com's Answer #689401 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Counterexample for (a) \r\n" );
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document.write( "⌊1.6*1.7⌋ ?≤? ⌊1.6⌋*⌊1.7⌋ \r\n" );
document.write( "⌊2.72⌋ ?≤? ⌊1⌋*⌊1⌋ \r\n" );
document.write( "2 ?≤? 1*1\r\n" );
document.write( "2 ?≤? 1\r\n" );
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document.write( "False, so (a) is disproved.\r\n" );
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document.write( "Proof for (b) \r\n" );
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document.write( "Let i = the integer part of x\r\n" );
document.write( "Let f = the fraction part of x\r\n" );
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document.write( "Let j = the integer part of y\r\n" );
document.write( "Let g = the fraction part of y\r\n" );
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document.write( "Then x = i+f, 0 < f < 1\r\n" );
document.write( "     y = j+g, 0 < g < 1\r\n" );
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document.write( "⌈x*y⌉ ?≤? ⌈x⌉*⌈y⌉\r\n" );
document.write( "⌈(i+f)*(j+g)⌉ ?≤? ⌈i+f⌉*⌈j+g⌉\r\n" );
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document.write( "⌈ij+ig+fj+fg⌉ ?≤? (i+1)*(j+1)\r\n" );
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document.write( "⌈ij+ig+fj+fg⌉ ?≤? ij+i+j+1\r\n" );
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document.write( "since ig < i and fj < j and fg < 1, that's true.\r\n" );
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document.write( "So the theorem is proved for positive real numbers. \r\n" );
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document.write( "[it's not true for negative real numbers though.]\r\n" );
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document.write( "Edwin
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