document.write( "Question 1203441: Which of the ff are associative binary operations? Show your solutions?
\n" ); document.write( "i. (ℤ,*), where x*y=(x+y)-(x·y) for all x,y,∈ℤ.
\n" ); document.write( "ii. (ℝ,*), where x*y=max (x,y) for all x,y,∈ℝ.
\n" ); document.write( "iii. (ℝ,*), where x*y= |x+y| for all x,y,∈ℝ. \n" ); document.write( "

Algebra.Com's Answer #839016 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
Which of the ff are associative binary operations? Show your solutions?
\n" ); document.write( "i. (ℤ,*), where x*y=(x+y)-(x·y) for all x,y,∈ℤ.
\n" ); document.write( "ii. (ℝ,*), where x*y=max (x,y) for all x,y,∈ℝ.
\n" ); document.write( "iii. (ℝ,*), where x*y= |x+y| for all x,y,∈ℝ.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " I will solve (ii) and (iii).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "(ii)   ((x*y)*z) = (max(x,y)*z) = obviously = max(x,y,z).\r\n" );
document.write( "\r\n" );
document.write( "       (x*(y,z)) = (x*max(y,z)) = obviously = max(x,y,z).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "       The expressions generate the same output for all x, y, z, so this operation (*) is associative.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(iii) Take  x= 1,  y= 2, z= -3.  Then\r\n" );
document.write( "\r\n" );
document.write( "      ((x*y)*z) = ||x+y| + z| = ||1+2| + (-3)| = |3 + (-3)| = |0| = 0.\r\n" );
document.write( "\r\n" );
document.write( "      (x*(y*z)) = |x + |y+z|| = |1 + |2+(-3)|| = |1 + |-1|| = |1 + 1| = |2| = 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "       The expressions generate different results for given x, y, z, so this operation (*) is NOT associative.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-----------------\r
\n" ); document.write( "\n" ); document.write( "

The lesson to learn is this:

\r
\n" ); document.write( "\n" ); document.write( " To prove that the operation is associative, the proof should be done for all x, y, z. \r
\n" ); document.write( "\n" ); document.write( " To prove that the operation is not associative, it is enough to show it using special counter-examples.
\n" ); document.write( "\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );