document.write( "Question 1042855: Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q. \n" ); document.write( "

Algebra.Com's Answer #657975 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Let a, b ∈ R. Prove using contradiction that if a ∈ Q and ab ∉ Q then b ∉ Q.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Let assume that b ∈ Q.   (1)\r\n" );
document.write( "\r\n" );
document.write( "Then  ab ∈ Q.\r\n" );
document.write( "\r\n" );
document.write( "But  ab ∉ Q, according to the condition.\r\n" );
document.write( "\r\n" );
document.write( "Contradiction.\r\n" );
document.write( "\r\n" );
document.write( "It means that the original assumption (1) is wrong.\r\n" );
document.write( "\r\n" );
document.write( "Hence,  b ∉ Q.\r\n" );
document.write( "\r\n" );
document.write( "QED.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Proved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "_______________
\n" ); document.write( "(*) Actually, it is usually considered as trivial and obvious statement . . . \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );