document.write( "Question 1088149: Dear Tutor, please help me.\r
\n" ); document.write( "\n" ); document.write( "Let A = {(1,2), (2,4), (3,6), (1,4), (2,8), (3,12), (1,3), (2,6), (3,4)}. Let r by the relation defined by (a,b)r(c,d) if and only if ad = bc.
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\n" ); document.write( "(a) The relation r is reflexive. Give one example of two elements of r (not A ) that demonstrate the reflexive property. Show clearly that the elements you choose satisfy the reflexive property.\r
\n" ); document.write( "\n" ); document.write( "(b) The relation r is symmetric. Give one example of two elements of r (not A) that demonstrate the symmetric property. Show clearly that the elements you choose satisfy the symmetric property.\r
\n" ); document.write( "\n" ); document.write( "(c) The relation r is transitive. Give one example of three elements of r (not A) that demonstrate the transitive property. Show clearly that the elements you choose satisfy the transitive property.\r
\n" ); document.write( "\n" ); document.write( "(d) As r is reflexive, symmetric and transitive it follows that r is an equivalence relation. What are the equivalence classes of r ?\r
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Algebra.Com's Answer #702422 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Let me to educate you a bit (a little).\r\n" );
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document.write( "        The equality ad = bc means that the determinant of the matrix  is equal to zero.        (*)\r\n" );
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document.write( "        This equality also means that the vectors (a,b) and (c,d) are proportional.                  (**)\r\n" );
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document.write( "    These  (*)  and  (**)  are EQUIVALENT properties.\r\n" );
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document.write( "Having this HINT will help you easily answer your questions.\r\n" );
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document.write( "a)  Regarding \"reflexivity\" property\r\n" );
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document.write( "    The property (the term) \"reflexive\" means that element E satisfies the relation ErE.\r\n" );
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document.write( "    ANY element of your list satisfies this property.\r\n" );
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document.write( "        By the way, to check whether this property is valid, you DO NOT NEED to treat two different elements of your set.\r\n" );
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document.write( "        The reflexive property must be checked for EACH element individually, not for the pair of elements.\r\n" );
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document.write( "        In this sense the formulation a) in your post IS INCORRECT.\r\n" );
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document.write( "b)  Regarding \"symmetric\" property\r\n" );
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document.write( "    The example of two elements that demonstrate the symmetric property is \r\n" );
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document.write( "    (1,2)  and  (2,4).\r\n" );
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document.write( "    You may check it immediately, making the necessary calculations manually.\r\n" );
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document.write( "c)  Regarding \"transitivity\" property.\r\n" );
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document.write( "    The property (**) makes transitivity OBVIOUS in this case:\r\n" );
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document.write( "         If vector A is proportional to vector B, And B is proportional to C, then CLEARLY A is proportional to C \r\n" );
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document.write( "         and the transitivity is on the place.\r\n" );
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document.write( "d)  Regarding classes of equivalency.\r\n" );
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document.write( "    The element (1,2) has equivalent (2,4), (3,6).                  (1)\r\n" );
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document.write( "    The element (1,4) has equivalent (2,8), (3,12).                 (2)\r\n" );
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document.write( "    The element (1,3) has equivalent (2,6).                         (3)\r\n" );
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document.write( "    The element (3,4) is equivalent to itself and to nothing else.  (4)\r\n" );
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document.write( "    So, the lists (1), (2), (3) and (4) represent four classes of equivalency.\r\n" );
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