document.write( "Question 1013595: it is question from set theory. I can't write the notations clearly because they are not programmed on my phone . so please bear me . the question goes follows: (A-B)UB=AUB. \n" ); document.write( "

Algebra.Com's Answer #629871 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "It is question from set theory. I can't write the notations clearly because they are not programmed on my phone. So please bear me. the question goes follows: (A-B)UB=AUB.
\n" ); document.write( "---------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "First, explanation.\r\n" );
document.write( "The set A-B is the set of those elements of the set A that do not belong to the set B.\r\n" );
document.write( "\r\n" );
document.write( "  It may happen that you just now what is A-B. \r\n" );
document.write( "  So, this explanation is just in case.\r\n" );
document.write( "\r\n" );
document.write( "Now let us prove that each element of the set (A-B)UB belongs to AUB.\r\n" );
document.write( "\r\n" );
document.write( "Actually, it is absolutely clear, because the set (A-B)UB is simply a subset of AUB.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next, let us prove that each element of the set AUB belongs to (A-B)UB.\r\n" );
document.write( "\r\n" );
document.write( "Let x belongs to AUB. \r\n" );
document.write( "Then x belongs to A or x belongs to B or x belongs to both A and B.\r\n" );
document.write( "In any case x belongs to (A-B) or B.\r\n" );
document.write( "\r\n" );
document.write( "Thus we proved that each element of the (A-B)UB belongs to AUB and , in opposite, each element of the AUB belongs to (A-B)UB. \r\n" );
document.write( "Hence, x belongs to (A-B)UB.\r\n" );
document.write( "\r\n" );
document.write( "Thus these sets consist of the same elements.\r\n" );
document.write( "Therefore, these sets are identical.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Is everything clear to you?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );