Groups: Math error done by student

For proving the questions on Groups many students commits mistakes on the following questions.
1. If Q1 is the set of rationall numbers other than 1 with binary operation * defined by a*b=a+b-ab for all a,b εQ1, Show that (Q1,*) is an abelian group and solve 5*x=3 in Q1.
2.If Q-1 is the set of all rational numbers except -1 and * is a binary operation defined on Q-1 by a*b= a+b+ab, Prove that Q-1 is an abelian group.
For verifying closure law for the above the problems, many students simply do in the following way
Wrong approach:
Closure law: a*b=a+b-ab is again a rational number and hence closure law is true.
However Here student did not verified a*b is a rational number except 1. Hence it is very necessary to show that a*b is a rational numbr other than 1 . Other wise 1Mark will be reduced in the valuation. In april 2002, the 1 Mark is reduced for the students who did not verified the closure law correctly.
The correct approach is as follows:
For question number 1:
Let a, b Q1 such that a #1, b#1 a+b-ab is also a rational number and it cannot be equal to 1
Because If a+b-ab=1, then a+b-ab-1=0
then (a-1)(b-1)=0
thereofore a=1 or b=1 which is not true as a#1 and b#1. a*b belongs to Q1
therefore Q1 is closed w.r.t * (GIVING REASONS IS IMP)
Similarly for verfying closure for 2 question
Closure law: or Q-1 a*b=a+b-ab is again a rational number and hence closure law is true.
Let a, b Q-1such that a # -1 b#-1 a+b+ab is also a rational number and it cannot be equal to -1
Because If a+b+ab=-1, then a+b+ab+1=0
therefore (a+1)(b+1)=0
Hence a= -1 or b= -1 which is not true as a# -1 and b# -1. a*b belongs to Q-1
Q-1 is closed w.r.t * (GIVING REASONS IS IMP)

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Comments

  • JAYARAM.K.H  On March 7, 2009 at 7:49 am

    You are doing good thing.I felt happy.

  • sachin bangera  On March 7, 2009 at 7:49 am

    very nice

  • VAISHNAVI  On March 7, 2009 at 3:14 pm

    thank u

  • Yashas  On March 6, 2010 at 9:20 am

    a very important approach which will surely save our precious one mark.
    thank you

  • Usmaan Farooq  On March 14, 2010 at 9:43 pm

    thank you……! its flit happy

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