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let R* be the set of all real numbers except 0. define * on R* by letting a*b = |a|b.
\n" ); document.write( "a) show that * gives an associative binary operation on R*.
\n" ); document.write( "TST (A*B)*C=A*(B*C)
\n" ); document.write( "LHS=(|A|B)*C=||A|B|C......WHETHER A AND B ARE +VE OR -VE ,THE FINAL ANSWR IS
\n" ); document.write( "|AB|C
\n" ); document.write( "RHS=A*(|B|C)=|A||B|C...HERE TOO THE FINAL ANSWER IS |AB|C
\n" ); document.write( "HENCE ASSOCIATIVE PROPERTY IS PROVED.
\n" ); document.write( "b)show that there is a left identity for * and a right inverse for each element in R*.
\n" ); document.write( "TST THERE IS X WHERE ....X*A=A
\n" ); document.write( "X*A=|X|A=A...THAT IS X=1 IS THE LEFT IDENTITY
\n" ); document.write( "BUT A*1=|A|1 IS NOT EQUAL TO A IF A IS -VE.HENCE X=1 IS ONLY LEFT IDENTITY AND NOT A RIGHT IDENTITY.
\n" ); document.write( "RIGHT INVERSE...LET A*A'=1
\n" ); document.write( "A*A'=|A|A'=1..SO A'=1/|A|WHICH EXISTS SINCE A IS NOT EQUAL TO ZERO AND IS INDEPENDENT OF SIGN OF A
\n" ); document.write( "LEFT INVERSE....A'*A=1=|A'|A=1....|A'|=1/A..WHICH EXISTS IF A IS POSITIVE.,BUT DOES NOT EXIST IF A IS NEGATIVE SINCE WE CANT FIND A' SUCH THAT |A'| IS NEGATIVE.
\n" ); document.write( "c)is R* with this binary operation a group?
\n" ); document.write( "WE HAVE TO PROVE
\n" ); document.write( "1.CLOSOURE....A*B IS AN ELEMENT IN R*...WHICH IS OK.SINCE A/B CANNOT BE ZEROS AS GIVEN
\n" ); document.write( "2.ASSOCIATIVE...PROVED ALREADY
\n" ); document.write( "3.EXISTENCE OF IDENTITY...WE HAVE 1*A=A...BUT A*1 IS NOT EQUAL TO A .HENCE THIS IS NOT A GROUP.
\n" ); document.write( "d) explain the significance of this exercise.
\n" ); document.write( "IT SHOWS THAT THERE COULD BE AN IDENTITY ELEMENT WHICH IS PARTIAL IN THE SENSE THAT IT IS ONLY LEFT IDENTITY AND NOT RIGHT IDENTITY. \n" ); document.write( "